• Richard K. Guy
Part of the Unsolved Problems in Intuitive Mathematics book series (PBM, volume 1)


We will denote by d(n) the number of positive divisors of n, by σ(n) the sum of those divisors, and by σ k (n) the sum of their kth powers, so that σ 0(n) = d(n) and σ 1(n) = σ(n). We use s(n) for the sum of the aliquot parts of n, i.e., the positive divisors of n other than n itself, so that s(n) = σ(n) − n.


Unsolved Problem Prime Divisor Numerical Math Acta Arith Consecutive Integer 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. M. Buxton and S. Elmore, An extension of lower bounds for odd perfect numbers, Notices Amer. Math. Soc. 23 (1976) A-55.Google Scholar
  2. E. Z. Chein, PhD thesis, Pennsylvania State Univ. 1979.Google Scholar
  3. E. Z. Chein, An odd perfect number has at least 8 prime factors, Abstract 79T-A 102, Notices Amer. Math. Soc. 26 (1979) A-365.Google Scholar
  4. G. L. Cohen, On odd perfect numbers, Fibonacci Quart. 16 (1978) 523–527; MR 80g:10010; Zbl. 391. 10008.Google Scholar
  5. Graeme L. Cohen, On odd perfect numbers (II), Multiperfect numbers and quasiperfect numbers, J. Austral. Math. Soc. Ser. A 29 (1980) 369–384.zbMATHGoogle Scholar
  6. G. L. Cohen and M. D. Hendy, Polygonal supports for sequences of primes, Math. Chronicle 9 (1980) 120–136.MathSciNetzbMATHGoogle Scholar
  7. J. T. Condict, On an odd perfect number’s largest prime divisor, senior thesis, Middlebury College, May, 1978.Google Scholar
  8. G. G. Dandapat, J. L. Hunsucker, and Carl Pomerance, Some new results on odd perfect numbers, Pacific J. Math. 57 (1975) 359–364.MathSciNetzbMATHCrossRefGoogle Scholar
  9. John A. Ewell, On the multiplicativestructure of odd perfect numbers, J. Number Theory 12 (1980) 339–342.Google Scholar
  10. O. Grün, Über ungerade vollkommene Zahlen, Math. Zeit. 55 (1952) 353–354.zbMATHCrossRefGoogle Scholar
  11. Peter Hagis, A lower bound for the set of odd perfect numbers, Math. Comput. 27 (1973) 951–953.zbMATHCrossRefGoogle Scholar
  12. P. Hagis, Every odd perfect number has at least 8 prime factors, Notices Amer. Math. Soc. 22 (1975) A-60.Google Scholar
  13. Peter Hagis, Outline of a proof that every odd perfect number has at least eight prime factors, Math. Comput. 34 (1980) 1027–1032.Google Scholar
  14. Peter Hagis, On the second largest prime factor of an odd perfect number. Proc. Grosswald Conf., Lecture Notes in Mathematics,Spring-Verlag, New York, (to be published).Google Scholar
  15. Peter Hagis and Wayne McDaniel, On the largest prime divisor of an odd perfect number, Math. Comput. 27 (1973) 955–957; MR 48 3855; Il, ibid. 29 (1975) 922–924.MathSciNetzbMATHGoogle Scholar
  16. H.-J. Kanold, Untersuchungen über ungerade vollkommene Zahlen J. reine angew. Math. 183 (1941) 98–109; MR 3 268.Google Scholar
  17. P. J. McCarthy, Odd perfect numbers, Scripta Math. 23 (1957) 43–47.MathSciNetGoogle Scholar
  18. Wayne McDaniel, On odd multiply perfect numbers, Boll. Un. Mat. Ital. (4) 3 (1970) 185–190; MR 41 6764.Google Scholar
  19. Wayne L. McDaniel and Peter Hagis, Some results concerning the non-existence of odd perfect numbers of the form p“M 213, Fibonacci Quart. 13 (1975) 25–28.MathSciNetzbMATHGoogle Scholar
  20. Joseph B. Muskat, On divisors of odd perfect numbers, Math. Comput. 20 (1966) 141–144; MR 32 4076.Google Scholar
  21. Karl K. Norton, Remarks on the number of factors of an odd perfect number, Acta Arith. 6 (1961) 365–374 (and see 36 (1980) 163); MR 26 4950.Google Scholar
  22. M. Perisastri, A note on odd perfect numbers, Math. Student 26 (1958) 179–181. C. Pomerance, Odd perfect numbers are divisible by at least seven distinct primes.Google Scholar
  23. Acta Arith.25 (1974) 265–300; see also Notices Amer. Math. Soc. 19 (1972) A-622–623.Google Scholar
  24. C. Pomerance, The second largest factor of an odd perfect number, Math. Comput. 29 (1975) 914–921.MathSciNetCrossRefGoogle Scholar
  25. Carl Pomerance, Multiply perfect numbers, Mersenne primes and effective computability, Math. Ann. 226 (1977) 195–206.MathSciNetzbMATHCrossRefGoogle Scholar
  26. N. Robbins, The non-existence of odd perfect numbers with less than seven distinct prime factors, PhD dissertation, Polytech. Inst. Brooklyn, June 1972. Hans Salié, Über abundante Zahlen, Math. Nachr. 9 (1953) 217–220.Google Scholar
  27. C. Servais, Sur les nombres parfaits, Mathesis 8 (1888) 92–93.Google Scholar
  28. Daniel Shanks, Solved and Unsolved Problems in Number Theory,2nd ed. Chelsea, New York 1978, esp. p. 217.Google Scholar
  29. Bryant Tuckerman, A search procedure and lower bound for odd perfect numbers, Math. Comput. 27 (1973) 943–949.zbMATHCrossRefGoogle Scholar
  30. H. Abbott, C. E. Aull, Ezra Brown, and D. Suryanarayana, Quasiperfect numbers, Acta Arith. 22 (1973) 439–447; MR 47 4915. Corrections, ibid. 29 (1976) 427–428.MathSciNetzbMATHGoogle Scholar
  31. M. M. Artuhov, On the problem of odd h-fold perfect numbers, Acta Arith. 23 (1973) 249–255.MathSciNetzbMATHGoogle Scholar
  32. P. T. Bateman, P. Erdös, C. Pomerance, and E. G. Straus, in Proc. Grosswald Conf., Springer-Verlag, New York, 1980.Google Scholar
  33. S. J. Benkoski, Problem E. 2308, Amer. Math. Monthly 79 (1972) 774.MathSciNetCrossRefGoogle Scholar
  34. S. J. Benkoski and P. Erdös, On weird and pseudoperfect numbers, Math. Comput.28 (1974) 617–623; MR 50 228 (Corrigendum, S. Kravitz, ibid. 29 (1975) 673 ).Google Scholar
  35. Alan L. Brown, Multiperfect numbers, Scripta Math. 20 (1954) 103–106; MR 16, 12.Google Scholar
  36. Paolo Catteneo, Sui numeri quasiperfetti, Boll. Un. Mat. Ital. (3) 6 (1951) 59–62; Zbl. 42, 268.Google Scholar
  37. Graeme L. Cohen, The non-existence of quasiperfect numbers of certain forms (to appear).Google Scholar
  38. G. L. Cohen and M. D. Hendy, On odd multiperfect numbers, Math. Chronicle 9 (1980). J. T. Cross, A note on almost perfect numbers, Math. Mag. 47 (1974) 230–231.MathSciNetCrossRefGoogle Scholar
  39. P. Erdös, Problems in number theory and combinatorics, Congressus Numerantium XVIII, Proc. 6th Conf. Numerical Math. Manitoba,1976, 35–58 (esp. pp. 53–54);Google Scholar
  40. Benito Franqui and Mariano Garcia, Some new multiply perfect numbers, Amer. Math. Monthly, 60 (1953) 459–462.MathSciNetzbMATHCrossRefGoogle Scholar
  41. Benito Franqui and Mariano Garcia, 57 new multiply perfect numbers, Scripta Math. 29 (1954) 169–171 (1955).MathSciNetGoogle Scholar
  42. Mariano Garcia, A generalization of multiply perfect numbers, Scripta Math. 19 (1953) 209–210.MathSciNetGoogle Scholar
  43. Mariano Garcia, On numbers with integral harmonic mean, Amer. Math. Monthly, 61 (1954) 89–96.MathSciNetzbMATHCrossRefGoogle Scholar
  44. P. Hagis and G. L. Cohen, Some results concerning quasiperfect numbers, J. Austral. Math. Soc.,(to appear).Google Scholar
  45. B. Hornfeck and E. Wirsing, Über die Häufigkeit vollkommener Zahlen, Math. Ann. 133 (1957) 431–438; MR 19, 837. See also ibid. 137 (1959) 316–318; MR 21 3389.Google Scholar
  46. R. P. Jerrard and Nicholas Temperley, Almost perfect numbers, Math. Mag. 46 (1973) 84–87.MathSciNetzbMATHCrossRefGoogle Scholar
  47. H.-J. Kanold, Über das harmonische Mittel der Teiler einer natürlichen Zahl, Math. Ann. 133 (1957) 371–374.MathSciNetCrossRefGoogle Scholar
  48. David G. Kendall, The scale of perfection, J. Appl. Probability 19A (P.A.P. Moran birthday volume, 1982.) (to appear)Google Scholar
  49. Masao Kishore, Odd almost perfect numbers, Notices Amer. Math. Soc. 22 (1975) A-380.Google Scholar
  50. Masao Kishore, Quasiperfect numbers are divisible by at least six distinct prime factors, Notices Amer. Math. Soc. 22 (1975) A-441.Google Scholar
  51. Masao Kishore, Odd integers N with 5 distinct prime factors for which 2–10 ’ 2 a(N)/N 2 + l0–12, Math. Comput. 32 (1978) 303–309.zbMATHGoogle Scholar
  52. M. S. Klamkin, Problem E.1445*, Amer. Math. Monthly 67 (1960) 1028. See also ibid. 82 (1975) 73.Google Scholar
  53. Sydney Kravitz, A search for large weird numbers, J. Recreational Math. 9 (1976–77) 82–85.Google Scholar
  54. A. kowski, Remarques sur les fonctions 0(n), ¢(n) et a(n), Mathesis 69 (1960) 302–303.MathSciNetGoogle Scholar
  55. A. M4kowski, Some equations involving the sum of divisors, Elem. Math. 34 (1979) 82; MR 81b: 10004.Google Scholar
  56. D. Minai, Structural issues for hyperperfect numbers, Fibonacci Quart.,(to appear).Google Scholar
  57. D. Minoli, Issues in non-linear hyperperfect numbers, Math. Comput. 34 (1980) 639645.Google Scholar
  58. Daniel Minoli and Robert Bear, Hyperperfect numbers, Pi Mu Epsilon J.6 (197475)153–157.Google Scholar
  59. Oystein Ore, On the averages of the divisors of a number, Amer. Math. Monthly 55 (1948) 615–619.MathSciNetCrossRefGoogle Scholar
  60. Seppo Pajunen, On primitive weird numbers, A collection of manuscripts related to the Fibonacci sequence,18th anniv. vol., Fibonacci Assoc., 162–166.Google Scholar
  61. Carl Pomerance, On a problem of Ore: harmonic numbers (unpublished typescript). Carl Pomerance, On multiply perfect numbers with a special property, Pacific J. Math. 57 (1975) 511–517.MathSciNetzbMATHCrossRefGoogle Scholar
  62. Carl Pomerance, On the congruences u(n) = a (mod n) and n = a (mod 0(n)), Acta Arith. 26 (1975) 265–272.zbMATHGoogle Scholar
  63. Paul Poulet, La Chasse aux Nombres, Fascicule I, Bruxelles, 1929, 9–27.zbMATHGoogle Scholar
  64. Problem B-6, William Lowell Putnam Mathematical Competition, 1976: 12: 04.Google Scholar
  65. Herman J. J. te Riele, Hyperperfect numbers with three different prime factors, Math. Comput. 36 (1981) 297–298.zbMATHCrossRefGoogle Scholar
  66. Neville Robbins, A class of solutions of the equation a(n) = 2n + t, Fibonacci Quart. 18 (1980) 137–147 (misprints in solutions for t = 31, 84, 86).Google Scholar
  67. H. N. Shapiro, Note on a theorem of Dickson, Bull. Amer. Math. Soc. 55 (1949) 450–52.MathSciNetzbMATHCrossRefGoogle Scholar
  68. H. N. Shapiro, On primitive abundant numbers, Comm. Pure Appl. Math. 21 (1968) 111–118.MathSciNetzbMATHCrossRefGoogle Scholar
  69. W. Sierpipski, Sur les nombres pseudoparfaits, Mat. Vesnik 2 (17) (1965) 212–213; MR 33 7296.Google Scholar
  70. W. Sierpipski, Number Theory (in Polish), 1959, p. 257.Google Scholar
  71. D. Suryanarayana, Quasi-perfect numbers II Bull. Calcutta Math. Soc. 69 (1977) 421–426; MR 80m:10003.Google Scholar
  72. Andreas and Eleni Zachariou, Perfect, semi-perfect and Ore numbers, Bull. Soc. Math. Grèce (N.S.) 13 (1972) 12–22; MR 50 1 2905.Google Scholar
  73. H. A. M. Frey, Über unitär perfekte Zahlen, Elem. Math. 33 (1978) 95–96; MR 81a: 10007.Google Scholar
  74. M. V. Subbarao, Are there an infinity of unitary perfect numbers?, Amer. Math. Monthly 77 (1970) 389–390.MathSciNetCrossRefGoogle Scholar
  75. M. V. Subbarao and D. Suryanarayana, Sums of the divisor and unitary divisor functions, J. reine angew. Math. 302 (1978) 1-15; MR 80d: 10069.Google Scholar
  76. M. V. Subbarao and L. J. Warren, Unitary perfect numbers, Canad. Math. Bull. 9 (1966) 147–153; MR 33 3994.Google Scholar
  77. M. V. Subbarao, T. J. Cook, R. S. Newberry, and J. M. Weber, On unitary perfect numbers, Delta 3 1 (Spring 1972 ) 22–26.Google Scholar
  78. Charles R. Wall, The fifth unitary perfect number, Canad. Math. Bull. 18 (1975)115–122. See also Notices Amer. Math. Soc. 16 (1969) 825.Google Scholar
  79. J. Alanen, O. Ore, and J. G. Stemple, Systematic computations on amicable numbers, Math. Comput. 21 (1967) 242–245; MR 36 5058.Google Scholar
  80. M. M. Artuhov, On some problems in the theory of amicable numbers (Russian), Acta Arith. 27 (1975) 281–291.MathSciNetGoogle Scholar
  81. W. Borho, On Thabit ibn Kurrah’s formula for amicable numbers, Math. Comput. 26 (1972) 571–578.MathSciNetzbMATHGoogle Scholar
  82. W. Borho, Befreundete Zahlen mit gegebener Primteileranzahl, Math. Ann. 209 (1974) 183–193.MathSciNetzbMATHCrossRefGoogle Scholar
  83. W. Borho, Eine Schranke für befreundete Zahlen mit gegebener Teileranzahl, Math. Nachr. 63 (1974) 297–301.MathSciNetzbMATHCrossRefGoogle Scholar
  84. W. Borho, Some large primes and amicable numbers, Math. Comput. 36 (1981) 303–304. P. Bratley and J. McKay, More amicable numbers, Math. Comput. 22 (1968) 677–678; MR 37 I299.Google Scholar
  85. P. Bratley, F. Lunnon, and J. McKay, Amicable numbers and their distribution, Math. Comput. 24 (1970) 431–432.MathSciNetzbMATHCrossRefGoogle Scholar
  86. B. H. Brown, A new pair of amicable numbers, Amer. Math. Monthly, 46 (1939) 345.zbMATHCrossRefGoogle Scholar
  87. Patrick Costello, Four new amicable pairs, Notices Amer. Math. Soc. 21 (1974) A-483.Google Scholar
  88. Patrick Costello, Amicable pairs of Euler’s first form, Notices Amer. Math. Soc. 22 1975) A-440.Google Scholar
  89. P. Erdös, On amicable numbers, Publ. Math. Debrecen 4 (1955) 108–111; MR 16, 998.Google Scholar
  90. P. Erdös and G. J. Rieger, Ein Nachtrag über befreundete Zahlen, J. reine angew. Math. 273 (1975) 220.MathSciNetzbMATHGoogle Scholar
  91. E. B. Escott, Amicable numbers, Scripta Math. 12 (1946) 61–72; MR 8, 135.Google Scholar
  92. M. Garcia, New amicable pairs, Scripta Math. 23 (1957) 167–171; MR 20 5158.Google Scholar
  93. A. A. Gioia and A. M. Vaidya, Amicable numbers with opposite parity, Amer. Math.Google Scholar
  94. Monthly 74 (1967) 969–973; correction 75 (1968) 386; MR 36 3711, 37 1306.Google Scholar
  95. P. Hagis, On relatively prime odd amicable numbers, Math. Comput. 23 (1969) 539–543; MR 40 85.Google Scholar
  96. P. Hagis, Lower bounds for relatively prime amicable numbers of opposite parity, ibid. 24 (1970) 963–968.MathSciNetzbMATHGoogle Scholar
  97. P. Hagis, Relatively prime amicable numbers of opposite parity, Math. Mag. 43 (1970) 14–20.MathSciNetzbMATHCrossRefGoogle Scholar
  98. H.-J. Kanold, Über die Dichten der Mengen der vollkommenen und der befreundeten Zahlen, Math. Z. 61 (1954) 180–185; MR 16, 337.Google Scholar
  99. H.-J. Kanold, Über befreundete Zahlen I, Math. Nachr. 9 (1953) 243–248; II, ibid. 10 (1953) 99–111; MR 15, 506.Google Scholar
  100. H.-J. Kanold, Über befreundete Zahlen III, J. reine angew. Math. 234 (1969) 207–215; MR 39 122.Google Scholar
  101. E. J. Lee, Amicable numbers and the bilinear diophantine equation, Math. Comput. 22 (1968) 181–187; MR 37 142.Google Scholar
  102. E. J. Lee, On divisibility by nine of the sums of even amicable pairs, Math. Comput. 23 (1969) 545–548; MR 40 1328.Google Scholar
  103. E. J. Lee and J. S. Madachy, The history and discovery of amicable numbers, part 1, J. Recreational Math. 5 (1972) 77–93; part 2, ibid. 153–173; part 3, ibid. 231–249.Google Scholar
  104. O. Ore, Number Theory and its History, McGraw-Hill, New York, 1948, p. 89. Carl Pomerance, On the distribution of amicable numbers, J. reine angew. Math. 293/294 (1977) 217–222; II ibid. (1981).Google Scholar
  105. P. Poulet, 43 new couples of amicable numbers, Scripta Math. 14 (1948) 77.zbMATHGoogle Scholar
  106. H. J. J. te Ride, Four large amicable pairs, Math. Comput. 28 (1974) 309–312.Google Scholar
  107. Walter E. Beck and Rudolph M. Najar, More reduced amicable pairs, Fibonacci Quart. 15 (1977) 331–332; Zbl. 389. 10004.Google Scholar
  108. Walter E. Beck and Rudolph M. Najar, Fixed points of certain arithmetic functions, Fibonacci Quart. 15 (1977) 337–342; Zbl. 389. 10005.Google Scholar
  109. Peter Hagis and Graham Lord, Quasi-amicable numbers, Math. Comput. 31 (1977) 608–611; MR 55 7902; Zbl. 355. 10010.Google Scholar
  110. M. Lal and A. Forbes, A note on Chowla’s function, Math. Comput. 25 (1971) 923–925; MR 45 6737; Zbl. 245. 10004.Google Scholar
  111. H. W. Lenstra has proved that it is possible to construct arbitrarily long monotonic increasing aliquot sequences.Google Scholar
  112. Jack Alanen, Empirical study of aliquot series, Math. Rep. 133, Stichting Math. Centrum, Amsterdam, 1972; reviewed Math. Comput. 28 (1974) 878–880.Google Scholar
  113. E. Catalan, Bull. Soc. Math. France 16 (1887–88) 128–129.Google Scholar
  114. John S. Devitt, Aliquot Sequences, MSc thesis, The Univ. of Calgary, 1976; see Math. Comput. 32 (1978) 942–943.Google Scholar
  115. J. S. Devitt, R. K. Guy, and J. L. Selfridge, Third report on aliquot sequences, Congressus Numeratium XVIII Proc. 6th Manitoba Conf. Numerical Math. 1976, 177–204; MR 80d:10001.Google Scholar
  116. L. E. Dickson, Theorems and tables on the sum of the divisors of a number, Quart. J. Math. 44 (1913) 264–296.zbMATHGoogle Scholar
  117. Paul Erdös, On asymptotic properties of aliquot sequences, Math. Comput. 30 (1976) 641–645.zbMATHGoogle Scholar
  118. Richard K. Guy, Aliquot sequences, in Number Theory and Algebra, Academic Press, 1977, 111–118; MR 57 223; Zbl. 367. 10007.Google Scholar
  119. Richard K. Guy and J. L. Selfridge, Interim report on aliquot series, Congressus Numerantium V, Proc. Conf. Numerical Math. Winnipeg, 1971, 557–580; MR 49 194; Zbl. 266. 10006.Google Scholar
  120. Richard K. Guy and J. L. Selfridge, Combined report on aliquot sequences, The Univ. of Calgary Math. Res. Report No. 225, May, 1974.Google Scholar
  121. Richard K. Guy and J. L. Selfridge, What drives an aliquot sequence? Math. Comput. 29 (1975) 101–107; MR 52 5542; Zbl. 296.10007. Corrigendum, ibid. 34 (1980).Google Scholar
  122. Richard K. Guy and M. R. Williams, Aliquot sequences near 1012, Congressus Numerantium XII, Proc. 4th Conf. Numerical Math. Winnipeg, 1974, 387–406; MR 52 242; Zbl. 359. 10007.Google Scholar
  123. Richard K. Guy, D. H. Lehmer, J. L. Selfridge, and M. C. Wunderlich, Second report on aliquot sequences, Congressus Numerantium IX, Proc. 3rd Conf. Numerical Math. Winnipeg, 1973, 357–368; MR 50 4455; Zbl. 325. 10007.Google Scholar
  124. G. Aaron Paxson, Aliquot sequences (preliminary report), Amer. Math. Monthly, 63 (1956) 614. See also Math. Comput. 26 (1972) 807–809.Google Scholar
  125. P. Poulet, La chasse aux nombres, Fascicule I, Bruxelles, 1929.zbMATHGoogle Scholar
  126. H. J. J. te Riele, A note on the Catalan—Dickson conjecture, Math. Comput. 27 (1973) 189–192; MR 48 3869; Zbl. 255. 10008.Google Scholar
  127. W. Borho, Über die Fixpunkte der k-fach iterierten Teilersummenfunktion, Mitt. Math. Gesellsch. Hamburg 4 (1969) 35–38; MR 40 ¢7189.Google Scholar
  128. H. Cohen, On amicable and sociable numbers, Math. Comput. 24 (1970) 423–429; MR 42 5887. ichard David, letter to D. H. Lehmer, 1972: 02: 25.Google Scholar
  129. P. Poulet, Question 4865, L’Intermédiaire des math. 25 (1918) 100–101.Google Scholar
  130. S. C. Root, in M. Beeler, R. W. Gosper and R. Schroeppel, M.I.T. Artificial Intelligence Memo 239, 1972: 02: 29.Google Scholar
  131. Paul Erdös, A mélange of simply posed conjectures with frustratingly elusive solutions, Math. Mag. 52 (1979) 67–70.MathSciNetzbMATHCrossRefGoogle Scholar
  132. P. Erdös, Problems and results in number theory and graph theory, Congressus Numerantium XXVII, Proc. 9th Manitoba Conf. Numerical Math. Comput. 1979, 3–21.Google Scholar
  133. Richard K. Guy and Marvin C. Wunderlich, Computing unitary aliquot sequences—a preliminary report, Congressus Numerantium XXVII, Proc. 9th Manitoba Conf. Numerical Math. Comput. 1979, 257–270.Google Scholar
  134. P. Hagis, Unitary amicable numbers, Math. Comput. 25 (1971) 915–918; MR 45 8599; Zbl. 232. 10004.Google Scholar
  135. Peter Hagis, Unitary hyperperfect numbers, Math. Comput. 36 (1981) 299–301. M. Lal, G. Tiller and T. Summers, Unitary sociable numbers, Congressus Numerantium VII, Proc. 2nd Conf. Numerical Math. Winnipeg, 1972, 211–216; R 50114471; Zbl. 309. 10005.Google Scholar
  136. H. J. J. te Riele, Unitary Aliquot Sequences, MR 139/72, Mathematisch Centrum, msterdam, 1972; reviewed Math. Comput. 32 (1978) 944–945; Zbl. 251. 10008.Google Scholar
  137. H. J. J. te Riele, Further Results on Unitary Aliquot Sequences, NW2/73, Mathematisch Centrum, Amsterdam, 1973; reviewed Math. Comput. 32 (1978) 945.Google Scholar
  138. H. J. J. te Riele, A Theoretical and Computational Study of Generalized Aliquot Sequences, MCT74, Mathematisch Centrum, Amsterdam, 1976; reviewed Math. Comput. 32 (1978) 945–946; MR 58 27716.Google Scholar
  139. C. R. Wall, Topics related to the sum of unitary divisors of an integer, PhD thesis, Univ. of Tennessee, 1970.Google Scholar
  140. Dieter Bode, Über eine Verallgemeinerung der Vollkommenen Zahlen, Dissertation, Braunschweig, 1971.Google Scholar
  141. P. Erdös, Some remarks on the iterates of the 4 and a functions, Colloq. Math. 17 (1967) 195 —202.Google Scholar
  142. J. L. Hunsucker and C. Pomerance, There are no odd superperfect numbers less than 7 x 10’, Indian J. Math. 17 (1975) 107–120.MathSciNetzbMATHGoogle Scholar
  143. H.-J. Kanold, Über “Super perfect numbers,” Elern. Math. 24 (1969) 61–62; MR 39 5463.Google Scholar
  144. Graham Lord, Even perfect and superperfect numbers, Elem. Math. 30 (1975) 87–88. A. Makowski and A Schinzel, On the functions i(n) and o(n), Colloq. Math. 13 (196465) 95–99.Google Scholar
  145. A. Schinzel, Ungelöste Probleme Nr. 30, Elem. Math. 14 (1959) 60–61.MathSciNetGoogle Scholar
  146. D. Suryanarayana, Super perfect numbers, Elem. Math. 24 (1969) 16–17; MR 39 5706. D. Suryanarayana, There is no odd superperfect number of the form p 2 a, Elem. Math. 28 (1973) 148–150.MathSciNetzbMATHGoogle Scholar
  147. P. Erdös, Über die Zahlen der Form 6(n) — n und n — On), Elem. Math. 28 (1973) 83–86.zbMATHGoogle Scholar
  148. Paul Erdös, Some unconventional problems in number theory, Astérisque 61 (1979) 73–82; Zbl. 399.10001; MR 81h: 10001.Google Scholar
  149. Richard K. Guy and Daniel Shanks, A constructed solution of v(n) = on + 1), Fibonacci Quart. 12 (1974) 299; MR 50 219; Zbl. 287. 10004.Google Scholar
  150. John L. Hunsucker, Jack Nebb, and Robert E. Stearns, Computational results concerning some equations involving Q(n), Math. Student 41 (1973) 285–289.MathSciNetGoogle Scholar
  151. A. Makowski, On some equations involving functions qp(n) and v(n), Amer. Math. Monthly 67 (1960) 668–70 correction, ibid. 68 (1971) 650.Google Scholar
  152. W.E. Mientka and R. L. Vogt, Computational results relating to problems concerning 0(n), Mat. Vesnik 7 (1970) 35–36.MathSciNetGoogle Scholar
  153. P. Erdös and M. Kac, Problem 4518, Amer. Math. Monthly 60 (1953) 47. Solution, R. Breusch, 61 (1954) 264–265.Google Scholar
  154. M. Sugunamma, PhD thesis, Sri Venkataswara Univ. 1969. P. Erdös, Problems and results on consecutive integers, Eureka 38 (1975–76) 3–8.Google Scholar
  155. P. Erdös and G. Szekeres, Über die Anzahl der Abelschen Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem, Acta Litt. Sci. Szeged 7 (1934) 95–102; Zbl. 10, 294.Google Scholar
  156. S. W. Golomb, Powerful numbers, Amer. Math. Monthly 77 (1970) 848–852; MR 42 1780.Google Scholar
  157. Andrzej Makowski, On a problem of Golomb on powerful numbers, Amer. Math. Monthly 79 (1972) 761.MathSciNetzbMATHCrossRefGoogle Scholar
  158. W. A. Sentance, Occurences of consecutive odd powerful numbers, Amer. Math. Monthly 88 (1981) 272–274.MathSciNetzbMATHCrossRefGoogle Scholar
  159. E. G. Straus and M. V. Subbarao, On exponential divisors, Duke Math. J. 41 (1974) 465–471; MR 50 2053.Google Scholar
  160. M. V. Subbarao, On some arithmetic convolutions, in The Theory of Arithmetic Functions, Springer-Verlag, New York, 1972.Google Scholar
  161. M. V. Subbarao and D. Suryanarayana, Exponentially perfect and unitary perfect numbers, Notices Amer. Math. Soc. 18 (1971) 798.Google Scholar
  162. P. Erdös and L. Mirsky, The distribution of values of the divisor function d(n), Proc. London Math. Soc. (3) 2 (1952) 257–271.zbMATHGoogle Scholar
  163. M. Nair and P. Shiu, On some results of Erdös and Mirsky, J. London Math. Soc. (2) 22 (1980) 197–203; and see ibid. 17 (1978) 228–230.Google Scholar
  164. A. Schinzel, Sur un problème concernant le nombre de diviseurs d’un nombre naturel, Bull. Acad. Polon. Sci. Ser. sci. math. astr. phys. 6 (1958) 165–167.MathSciNetzbMATHGoogle Scholar
  165. A. Schinzel and W. Sierpinski, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958) 185–208.MathSciNetzbMATHGoogle Scholar
  166. W. Sierpinski, Sur une question concernant le nombre de diviseurs premiers d’un nombre naturel, Colloq. Math. 6 (1958) 209–210.MathSciNetzbMATHGoogle Scholar
  167. A. Makowski, On a problem of Erdös, Enseignement Math. (2) 14 (1968) 193.MathSciNetzbMATHGoogle Scholar
  168. Robert Baillie, G. V. Cormack, and H. C. Williams, Some results concerning a problem of Sierpinski, Math. Comput.,(to appear).Google Scholar
  169. G. V. Cormack and H. C. Williams, Some very large primes of the form k 2“ + 1, Math. Comput. 35 (1980) 1419–1421; MR 81i: 10011.Google Scholar
  170. P. Erdös and A. M. Odlyzko, On the density of odd integers of the form (p - 1)2-“ and related questions, J. Number Theory 11 (1979) 257–263.MathSciNetzbMATHCrossRefGoogle Scholar
  171. J. L. Selfridge, Solution to problem 4995, Amer. Math. Monthly 70 (1963) 101.MathSciNetCrossRefGoogle Scholar
  172. W. Sierpinski, Sur un problème concernant les nombres k 2“ + 1, Elem. Math. 15 (1960) 73–74;MathSciNetzbMATHGoogle Scholar
  173. W. Sierpinski, 250 Problems in Elementary Number Theory, Elsevier, New York, 1970, 10, 64.Google Scholar
  174. K. Alladi and C. Grinstead, On the decomposition of n! into prime powers, J. Number Theory 9 (1977) 452–458.MathSciNetzbMATHCrossRefGoogle Scholar
  175. P. Erdös, Some problems in number theory, Computers in Number Theory, Academic Press, London and New York, 1971, 405–414.Google Scholar
  176. E. Ecklund and R. EggletonPrime factors of consecutive integers Amer. Math. Monthly 79 (1972) 1082–1089.Google Scholar
  177. E. Ecklund, R. Eggleton, P. Erdös and J. L. Selfridge, On the prime factorization of binomial coefficients, J. Austral Math. Soc. Ser. A, 26 (1978) 257–269; MR 80e: 10009.Google Scholar
  178. P. Erdös, Problems and results on number theoretic properties of consecutive integers and related questions. Congressus Numerantium XVI Proc. 5th Manitoba Conf. Numer. Math. 1975, 25–44.Google Scholar
  179. P. Erdös and R. L. GrahamOn products of factorials Bull. Inst. Math. Acad. Sinica Taiwan 4 (1976) 337–355.Google Scholar
  180. Kenneth Lebensold, A divisibility problem, Studies in Applied Math. 56 (1976–77) 291–294. MR 58 2 1639.Google Scholar
  181. P. T. Bateman and R. M. Stemmler, Waring’s problem for algebraic number fields and primes of the form (p’ — 1)/(p 1 — 1), Illinois T. Math. 6 (1962) 142–156.Google Scholar
  182. Ted Chinburg and Melvin Henriksen, Sums of kth powers in the ring of polynomials with integer coefficients, Bull. Amer. Math. Soc. 81 (1975) 107–110.MathSciNetzbMATHCrossRefGoogle Scholar
  183. A. Makowski and A. Schinzel, Sur l’équation indéterminée de R. Goormaghtigh, Mathesis 68 (1959) 128–142 and 70 (1965) 94–96.zbMATHGoogle Scholar
  184. N. M. Stephens, On the Feit-Thompson conjecture, Math. Comput. 25 (1971) 625.MathSciNetzbMATHCrossRefGoogle Scholar
  185. S. L. G. ChoiOn sequences containing at most three pairwise coprime integers Trans. Amer. Math. Soc. 183 (1973) 437–440; MR 48 6052.Google Scholar
  186. P. ErdösExtremal problems in number theory Proc. Sympos. Pure Math. Amer. Math. Soc. 8 (1965) 181–189; MR 30 4740.Google Scholar
  187. P. Erdös and J. L. SelfridgeSome problems on the prime factors of consecutive integers Illinois J. Math. 11 (1967) 428–430.Google Scholar
  188. A. SchinzelUnsolved problem 31 Elem. Math. 14 (1959) 82–83.Google Scholar
  189. Alfred Brauer, On a property of k consecutive integers Bull. Amer. Math. Soc. 47 (1941) 328–331; MR 2, 248.Google Scholar
  190. Ronald J. EvansOn blocks of N consecutive integers Amer. Math. Monthly 76 (1969) 48–49.Google Scholar
  191. Ronald EvansOn N consecutive integers in an arithmetic progression Acta Sei. Math. Univ. Szeged 33 (1972) 295–296.Google Scholar
  192. Heiko Harborth, Eine Eigenschaft aufeinanderfolgender Zahlen, Arch. Math. (Basel) 21 (1970) 50–51.Google Scholar
  193. Heiko Harborth, Sequenzen ganzer Zahlen, in Zahlentheorie, Berichte aus dem Math. Forschungsinst. Oberwolfach 5 (1971) 59–66.Google Scholar
  194. S. S. Pillai, On m consecutive integers I, Proc. Indian Acad. Sci. Sect A, 11 (1940) 6–12; MR1,199; II ibid. 11 (1940) 73–80; MR 1,291; III ibid. 13 (1941) 530–533; MR 3, 66; IV Bull. Calcutta Math. Soc. 36 (1944) 99–101; MR 6, 170.Google Scholar
  195. D. H. LehmerOn a problem of Stormer Illinois J. Math. 8 (1964) 57–79; Google Scholar
  196. P. Erdös and G. Szekeres, Some number theoretic problems on binomial coefficients, Austral. Math. Soc. Gaz. 5 (1978) 97–99; MR 80e: 10010.Google Scholar
  197. P. Erdös, Problems and results in combinatorial analysis and combinatorial number theory, Congressus Numerantium XXI in Proc. 9th S.E. Conf. Combin. Graph Theory, Comput., Boca Raton, Utilitas Math. Winnipeg, 1978, 29–40.Google Scholar
  198. P. Erdös and C. Pomerance, Matching the natural numbers up to n with distinct multiples in another interval, Nederl. Akad. Wetensch. Proc. Ser. A 83 (=Indag. Math. 42) (1980) 147–161.Google Scholar
  199. Paul Erdös and Carl Pomerance, An analogue of Grimm’s problem of finding distinct prime factors of consecutive integers, Utilitas Math. 19 (1981), (to appear).Google Scholar
  200. P. Erdös and J. L. Selfridge, Some problems on the prime factors of consecutive integers II, in Proc. Washington State Univ. Conf. Number Theory, Pullman, 1971, 13–21.Google Scholar
  201. C. A. Grimm, A conjecture on consecutive composite numbers, Amer. Math. Monthly 76 (1969) 1126–1128.MathSciNetzbMATHCrossRefGoogle Scholar
  202. Michel Langevin, Plus grand facteur premier d’entiers en progression arithmétique, Sém. Delange-Pisot-Poitou, 18 (1976/77) Théorie des nombres, Fasc. 1, Exp. No. 3 (1977); MR 81a:10011.Google Scholar
  203. Carl Pomerance, Some number theoretic matching problems, in Proc. Number Theory Conf:, Queens Univ., Kingston, 1979, 237–247.Google Scholar
  204. K. T. Ramachandra, N. Shorey, and R. Tijdeman, On Grimm’s problem relating to factorization of a block of consecutive integers, J. reine angew. Math. 273 (1975) 109–124.MathSciNetzbMATHGoogle Scholar
  205. E. F. Ecklund, On prime divisors of the binomial coefficient, Pacific J. Math. 29 (1969) 267–270.MathSciNetzbMATHCrossRefGoogle Scholar
  206. P. Erdös, A theorem of Sylvester and Schur, J. London Math. Soc. 9 (1934) 282–288. Paul Erdös, A mélange of simply posed conjectures with frustratingly elusive solutions, Math. Mag. 52 (1979) 67–70.MathSciNetzbMATHCrossRefGoogle Scholar
  207. P. Erdös and R. L. Graham, On the prime factors of (z), Fibonacci Quart. 14 (1976) 348–352.MathSciNetzbMATHGoogle Scholar
  208. P. Erdös, R. L. Graham, I. Z. Ruzsa, and E. Straus, On the prime factors of („’), Math. Comput. 29 (1975) 83–92.zbMATHGoogle Scholar
  209. M. Faulkner, On a theorem of Sylvester and Schur, J. London Math. Soc. 41 (1966) 107–110.MathSciNetzbMATHCrossRefGoogle Scholar
  210. L. Moser, Insolvability of („„) _ (2:)(21,b), Canad. Math. Bull. 6 (1963) 167–169.MathSciNetzbMATHCrossRefGoogle Scholar
  211. I. Schur, Einige Sätze über Primzahlen mit Anwendungen and Irreduzibilitätsfragen I, S.-B. Preuss. Akad. Wiss. Phys.-Math. Kl. 14 (1929)125–136.Google Scholar
  212. J. Sylvester, On arithmetical series, Messenger of Math. 21 (1892) 1–19, 87–120. W. Utz, A conjecture of Erdös concerning consecutive integers, Amer. Math. Monthly 68 (1961) 896–897.MathSciNetCrossRefGoogle Scholar
  213. A. Schinzel, Sur un problème de P. Erdös, Colloq. Math. 5 (1957–58) 198–204.Google Scholar
  214. P. Erdös, How many pairs of products of consecutive integers have the same prime factors? Amer. Math. Monthly 87 (1980) 391–392.zbMATHCrossRefGoogle Scholar
  215. P. Erdös, Über die Zahlen der Form o(n)–n and n–4(n), Elem. Math. 28 (1973) 83–86.MathSciNetzbMATHGoogle Scholar
  216. P. Erdös and R. R. Hall, Distinct values of Euler’s (/)-function, Mathematika 23 (1976) 1–3.MathSciNetzbMATHCrossRefGoogle Scholar
  217. Andrzej Makowski, On the equation i(n + k) = 20(n), Elem. Math. 29 (1974) 13.Google Scholar
  218. A. Schinzel, Sur l’équation 4(x + k) _ q5(x), Acta Arith. 4 (1958) 181–184; MR 21 5597.Google Scholar
  219. A. Schinzel and A. Wakulicz, Sur l’équation c (x + k) _ 0(x) II, Acta Arith. 5 (1959) 425–426; MR 23 A831.Google Scholar
  220. W. Sierpinski, Sur un propriété de la fonction 0(n), Publ. Math. Debrecen 4 (1956) 184–185.MathSciNetGoogle Scholar
  221. Ronald Alter, Can 4(n) properly divide n — 1? Amer. Math. Monthly 80 (1973) 192–193. Graeme L. Cohen and Peter Hagis, On the number of prime factors of n if 4o(n)1(n — 1), Nieuw Arch. Wisk. (3) 28 (1980), 177–185.Google Scholar
  222. Masao Kishore, On the equation 0(M) = M — 1 Nieuw Arch. Wisk. (3) 25 (1977) 48–53. See also Notices Amer. Math. Soc. 22 (1975) A-501–502.Google Scholar
  223. D. H. Lehmer, On Euler’s totient function Bull. Amer. Math. Soc. 38 (1932) 745–751.Google Scholar
  224. E. Lieuwens, Do there exist composite numbers for which kO(M) = M — 1 holds? Nieuw Arch. Wisk. (3) 18 (1970) 165–169; MR 42 1750.Google Scholar
  225. C. Pomerance, On composite n for which i(n)In — 1, Acta Arith. 28 (1976) 387–389; II Pacific J. Math. 69 (1977) 177–186. See also Notices Amer. Math. Soc. 22 (1975) A-542.Google Scholar
  226. Fred Schuh, Can n — 1 be divisible by ¢(n) where n is composite? Mathematica Zutphen B. 12 (1944) 102–107.Google Scholar
  227. M. V. Subbarao, On two congruences for primality Pacific J. Math. 52 (1974) 261–268; MR 50 2049.Google Scholar
  228. David W. Wall, Conditions for 4o(N) to properly divide N — 1, A Collection of Manuscripts Related to the Fibonacci Sequence,18th Anniv. Vol., Fibonacci Assoc. 205–208.Google Scholar
  229. R. D. Carmichael, Note on Euler’s 4i-function, Bull. Amer. Math. Soc. 28 (1922) 109–110.MathSciNetzbMATHCrossRefGoogle Scholar
  230. P. Erdös, on the normal number of prime factors ofp 1 and some other related problems concerning Euler’s 0-function, Quart. J. Math. Oxford Ser. 6 (1935) 205–213.zbMATHCrossRefGoogle Scholar
  231. P. Erdös, Some remarks on Euler’s 0-function and some related problems, Bull. Amer. Math. Soc. 51 (1945) 540–544.MathSciNetzbMATHCrossRefGoogle Scholar
  232. P. Erdös, Some remarks on Euler’s 0 function, Acta Arith. 4 (1958)10–19; MR 22 1539. P. Erdös and R. R. Hall. Distinct values of Euler’s 0-function, Mathematika 23 (1976) 1–3.MathSciNetzbMATHCrossRefGoogle Scholar
  233. C. Hooley, On the greatest prime factor of p + a, Mathematika 20 (1973) 135–143. H. Iwaniec, On the Brun-Titchmarsh theorem, (to appear)Google Scholar
  234. V. L. Klee, On a conjecture of Carmichael, Bull. Amer. Math. Soc. 53 (1947)1183–1186; MR 9, 269.Google Scholar
  235. V. L. Klee, Is there an n for which 0(x) = n has a unique solution? Amer. Math. Monthly 76 (1969) 288–289.MathSciNetCrossRefGoogle Scholar
  236. Carl Pomerance, On Carmichael’s conjecture, Proc. Amer. Math. Soc. 43 (1974) 297–298.MathSciNetzbMATHGoogle Scholar
  237. Carl Pomerance, Popular values of Euler’s function, Mathematika 27 (1980) 84–89. K. R. Wooldridge, Values taken many times by Euler’s phi-function, Proc. Amer. Math. Soc. 76 (1979) 229–234; MR 80g: 10008.Google Scholar
  238. P. Erdös, On the integers relatively prime to n and on a number-theoretic function considered by Jacobsthal, Math. Scand. 10 (1962) 163–170; MR 26 3651.Google Scholar
  239. C. Hooley, On the difference of consecutive numbers prime to n, Acta Arith. 8 (1963) 343–347.MathSciNetzbMATHGoogle Scholar
  240. R. C. Vaughan, Some applications of Montgomery’s sieve, J. Number Theory 5 (1973) 64–79.MathSciNetzbMATHCrossRefGoogle Scholar
  241. P. A. Catlin, Concerning the iterated 0-function, Amer. Math. Monthly 77 (1970) 60–61. Paul Erdös and R. R. Hall, Euler’s 0-function and its iterates, Mathematika 24 (1977) 173–177; MR 57 1 2356.Google Scholar
  242. W. H. Mills, Iteration of the 0-function, Amer. Math. Monthly 50 (1953) 547–549.CrossRefGoogle Scholar
  243. C. A. Nicol, Some diophantine equations involving arithmetic functions, J. Math. Anal. Appl. 15 (1966) 154–161.MathSciNetzbMATHCrossRefGoogle Scholar
  244. Harold N. Shapiro, An arithmetic function arising from the 0-function, Amer. Math. Monthly 50 (1943) 18–30; MR 4, 188.Google Scholar
  245. A. Makowski and A. Schinzel, On the functions c (n) and a(n), Colloq. Math. 13 (196465) 95–99.Google Scholar
  246. L. Carlitz, A note on the left factorial function, Math. Balkanica 5 (1975) 37–42. Duro Kurepa, On some new left factorial propositions, Math. Balkanica 4 (1974) 383–386; MR 58 10716.Google Scholar
  247. E. Lehmer, On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson, Annals of Math. 39 (1938) 350–360; Zbl. 19, 5.Google Scholar
  248. Barry J. Powell, Advanced problem 6325, Amer. Math. Monthly 87 (1980) 826.MathSciNetCrossRefGoogle Scholar
  249. P. Erdös and Carl Pomerance, On the largest prime factors of n and n + 1, Aeguationes Math. 17 (1978) 311–321.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1981

Authors and Affiliations

  • Richard K. Guy
    • 1
  1. 1.Department of Mathematics and StatisticsThe University of CalgaryCalgaryCanada

Personalised recommendations