Sequences of Integers

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

Abstract

Here we are mainly, but not entirely, concerned with infinite sequences; there is some overlap with sections C and A. An excellent text and source of problems is H. Halberstam and K. F. Roth, Sequences, Vol. I, Oxford Univ. Press, 1966. It is to be hoped that Vol. II will follow in a finite time.

Keywords

Number Theory Greedy Algorithm Unsolved Problem Arithmetic Progression Infinite Sequence 
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.

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References

  1. P. Erdös, A. Sârközi, and E. Szemerédi, On divisibility properties of sequences of integers, in Number Theory, Colloq. Math. Soc. Janos Bolyai 2, North-Holland, 1970, 35–49.Google Scholar
  2. H. Ostmann, Additive Zahlentheorie I, II, Springer-Verlag, Heidelberg, 1956.MATHGoogle Scholar
  3. A. Stöhr, Gelöste und ungelöste Fragen über Basen der natürlichen Zahlenreihe, I, IIGoogle Scholar
  4. J. reine angew. Math. 194 (1955) 40–65, 111–140.Google Scholar
  5. L. Moser, On the additive completion of sets of integers, Proc. Symp. Pure Math. 8 Amer. Math. Soc. Providence, 1965, 175–180.Google Scholar
  6. I. Ruzsa, On a problem of P. Erdös, Canad. Math. Bull. 15 (1972) 309–310.MathSciNetMATHCrossRefGoogle Scholar
  7. P. Erdös, Problem, Mat. Lapok 2 (1951) 233.Google Scholar
  8. P. Erdös, On the density of some sequences of integers, Bull. Amer. Math. Soc. 54 (1948) 685–692; MR 10, 105.Google Scholar
  9. P. Erdös, On sequences of integers no one of which divides the product of two others and on some related problems, Inst. Math. Mec. Tomsk 2 (1938) 74–82.Google Scholar
  10. P. Erdös, Extremal problems in number theory V (Hungarian), Mat. Lapok 17 (1966) 135–155.MathSciNetMATHGoogle Scholar
  11. P. Erdös, On some applications of graph theory to number theory, Publ. Ramanujan Inst. 1 (1969) 131–136.Google Scholar
  12. A. S. Besicovitch, On the density of certain sequences, Math. Ann. 110 (1934) 335–341.Google Scholar
  13. P. Erdös, Note on sequences of integers no one of which is divisible by any other, J. London Math. Soc. 10 (1935) 126–128.CrossRefGoogle Scholar
  14. H. L. Abbott and A. C. Liu, On partitioning integers into progression free sets, J. Combin. Theory 13 (1972) 432–436.MathSciNetMATHCrossRefGoogle Scholar
  15. H. L. Abbott, A. C. Liu and J. Riddell, On sets of integers not containing arithmetic progressions of prescribed length J. Austral. Math. Soc. 18 (1974) 188–193; MR 57 #12441.Google Scholar
  16. Michael D. Beeler and Patrick E. O’Neil, Some new van der Waerden numbers, Discrete Math. 28 (1979) 135–146.MathSciNetMATHCrossRefGoogle Scholar
  17. F. A. Behrend, On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci. USA 32 (1946) 331–332; MR 8, 317.Google Scholar
  18. E. R. Berlekamp, A construction for partitions which avoid long arithmetic progressions, Canad. Math. Bull. 11 (1968) 409–414.MathSciNetMATHCrossRefGoogle Scholar
  19. E. R. Berlekamp, On sets of ternary vectors whose only linear dependencies involve an odd number of vectors, Canad. Math. Bull. 13 (1970) 363–366.MathSciNetMATHCrossRefGoogle Scholar
  20. Thomas C. Brown, Some new Van der Waerden numbers, Notices Amer. Math. Soc. 21(1974) A-432.Google Scholar
  21. T. C. Brown, Behrend’s theorem for sequences containing no k-element progression of a certain type, J. Combin. Theory Ser. A, 18 (1975) 352–356.MATHGoogle Scholar
  22. Ashok K. Chandra, On the solution of Moser’s problem in four dimensions, Canad. Math. Bull. 16 (1973) 507–511.MATHCrossRefGoogle Scholar
  23. V. Chvâtal, Some unknown van der Waerden numbers, in Combinatorial Structures and and their Applications, Gordon and Breach, New York, 1970, 31–33.Google Scholar
  24. V. Chvâtal, Remarks on a problem of Moser, Canad. Math. Bull. 15 (1972) 19–21.MathSciNetMATHCrossRefGoogle Scholar
  25. J. A. Davis, Roger C. Entringer, Ronald L. Graham and G. J Simmons, On permutations containing no long arithmetic progressions, Acta Arith. 34 (1977/78) 81–90; MR 58 #10705.Google Scholar
  26. P. Erdös, Some recent advances and current problems in number theory, in Lectures on Modern Mathematics, Wiley, New York, 3 (1965) 196–244.Google Scholar
  27. P. Erdös and R. Rado, Combinatorial theorems on classifications of subsets of a given set, Proc. London Math. Soc. (3) 2 (1952) 417–439; MR 16, 445.Google Scholar
  28. P. Erdös and J. Spencer, Probabilistic Methods in Combinatorics, Academic Press, 1974, 37–39.Google Scholar
  29. P. Erdös and P. Turân, On some sequences of integers, J. London Math. Soc. 11 (1936) 261–264.CrossRefGoogle Scholar
  30. F. Everts, PhD thesis, University of Colorado, 1977.Google Scholar
  31. H. Furstenberg, Ergodic behaviour of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math. 31 (1977) 204–256; MR 58 #16583.Google Scholar
  32. Joseph L. Gerver and Thomas L. Ramsey, Sets of integers with nonlong arithmetic progressions generated by the greedy algorithm, Math. Comput. 33 (1979) 1353–1359; MR 80k: 10053.Google Scholar
  33. R. L. Graham and B. L. Rothschild, A survey of finite Ramsey Theorems, Congressus Numerantium III, Proc. 2nd Louisiana Conf. Combin., Graph Theory, Comput. (1971) 21–40.Google Scholar
  34. R. L. Graham and B. L. Rothschild, A short proof of van der Waerden’s theorem on arithmetic progressions, Proc. Amer. Math. Soc. 42 (1974) 385–386.MathSciNetMATHGoogle Scholar
  35. G. Hajôs, Über einfache und mehrfache Bedeckung des n-dimensionalen Raumes mit einem Würfelgitter, Math. Z. 47 (1942) 427–467.CrossRefGoogle Scholar
  36. A. W. Hales and R. I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963) 222–229.MathSciNetMATHCrossRefGoogle Scholar
  37. A. Y. Khinchin, Three Pearls of Number Theory, Graylock Press, Rochester, N.Y. 1952, 11–17.MATHGoogle Scholar
  38. L. Moser, On non averaging sets of integers, Canad. J. Math. 5 (1953) 245–252.MATHCrossRefGoogle Scholar
  39. Leo Moser, Notes on number theory II. On a theorem of van der Waerden, Canad. Math. Bull. 3 (1960) 23–25; MR 22 #5619.Google Scholar
  40. L. Moser, Problem 21, Proc. Number Theory Conf. Univ. of Colorado, Boulder, 1963, 79.Google Scholar
  41. L. Moser, Problem 170, Canad. Math. Bull. 13 (1970) 268.Google Scholar
  42. A. M. Odlyzko and R. P. Stanley, Some curious sequences constructed with the greedy algorithm, Bell. Labs. internal memo. 1978.Google Scholar
  43. Carl Pomerance, Collinear subsets of lattice point sequences-an analog of Szemerédi’s theorem, J. Combin. Theory Ser. A. 25 (1980) 140–149.CrossRefGoogle Scholar
  44. John R. Rabung, On applications of van der Waerden’s theorem, Math. Mag. 48 (1975) 142–148.MathSciNetMATHCrossRefGoogle Scholar
  45. John R. Rabung, Some progression-free partitions constructed using Folkman’s method, Canad. Math. Bull. 22 (1979) 87–91.MathSciNetMATHCrossRefGoogle Scholar
  46. R. Rado, Note on combinatorial analysis, Proc. London Math. Soc. 48 (1945) 122–160.MathSciNetCrossRefGoogle Scholar
  47. R. A. Rankin, Sets of integers containing not more than a given number of terms in arithmetical progression, Proc. Roy. Soc. Edinburgh Sect. A 65 (1960/61) 332–334; MR 26 #95.Google Scholar
  48. J. Riddell, On sets of numbers containing no 1 terms in arithmetic progressions, Nieuw Arch. Wisk. (3) 17 (1969) 204–209; MR 41 #1678.Google Scholar
  49. K. F. Roth, Sur quelques ensembles d’entiers, C.R. Acad. Sci. Paris 234 (1952) 388–390.MathSciNetMATHGoogle Scholar
  50. K. F. Roth, On certain sets of integers, J. London Math. Soc. 28 (1953)104–109; MR 14, 536 (and see ibid. 29 (1954) 20–26MATHGoogle Scholar
  51. J. Number Theory 2 (1970) 125–142; Period. Math. Hungar. 2 (1972) 301–326 ).Google Scholar
  52. R. Salem and D. C. Spencer, On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci. USA 28 (1942) 561–563; MR 4, 131.Google Scholar
  53. R. Salem and D. C. Spencer, On sets which do not contain a given number in arithmetical progression, Nieuw Arch. Wisk. (2) 23 (1950) 133–143.MathSciNetMATHGoogle Scholar
  54. H. Salié, Zur Verteilung natürlicher Zahlen auf Elementfremde Klassen, Ber. Verh. Sächs. Akad. Wiss. Leipzig 4 (1954) 2–26.Google Scholar
  55. Wolfgang M. Schmidt, Two combinatorial theorems on arithmetic progressions, Duke Math. J. 29 (1962) 129–140.MathSciNetMATHCrossRefGoogle Scholar
  56. G. J. Simmons and H. L. Abbott, How many 3-term arithmetic progressions can there be if there are no longer ones? Amer. Math. Monthly 84 (1977) 633–635; MR 57 #3056.Google Scholar
  57. R. S. Stevens and R. Shantaram, Computer generated van der Waerden partitions, Math. Comput. 32 (1978) 635–636.MathSciNetMATHCrossRefGoogle Scholar
  58. E. Szemerédi, On sets of integers containing no four terms in arithmetic progression, Acta Math. Acad. Sci. Hungar. 20 (1969) 89–104.MathSciNetMATHCrossRefGoogle Scholar
  59. E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975) 199–245.MathSciNetMATHGoogle Scholar
  60. J. P. Thouvenot, La démonstration de Furstenberg du théorème de Szemerédi sur les progressions arithmétiques, Springer Lect. Notes in Math. 710, Berlin, 1979, 221–232; MR 81c: 10072.Google Scholar
  61. B. L. van der Waerden, Beweis einer Baudet’schen Vermutung, Nieuw Arch. voor Wisk. 11 15 (1927) 212–216.Google Scholar
  62. B. L. van der Waerden, How the proof of Baudet’s conjecture was found, in Studies in Pure Mathematics, Academic Press, London, 1971, 251–260.Google Scholar
  63. E. Witt, Ein kombinatorische Satz der Elementargeometrie, Math. Nachr. 6 (1952) 261–262.MathSciNetMATHCrossRefGoogle Scholar
  64. H. L. Abbott, PhD thesis, University of Alberta, 1965.Google Scholar
  65. H. L. Abbott and D. Hanson, A problem of Schur and its generalizations, Acta Arith. 20 (1972) 175–187.MathSciNetMATHGoogle Scholar
  66. H. L. Abbott and L. Moser, Sum-free sets of integers, Acta Arith. 11 (1966) 393–396; MR 34#69.Google Scholar
  67. L. D. Baumert, Sum-free sets, J. P. L. Res Summary No 36–10,1(1961)16–18.Google Scholar
  68. S. L. G. Choi, The largest sum-free subsequence from a sequence of n numbers, Proc. Amer. Math. Soc. 39 (1973) 42–44; MR 47 #1771.Google Scholar
  69. S. L. G. Choi, J. Komlas, and E. Szemerédi, On sum-free subsequences, Trans. Amer. Math. Soc. 212 (1975) 307–313.MathSciNetMATHCrossRefGoogle Scholar
  70. H. Fredericksen. Five sum-free sets, Proc. 6th Ann. S.E. Conf. Graph Theory, Combin. & Comput. Congressus Numerantium XIV, Utilitas Math. Pub. Inc 1975, 309–314.Google Scholar
  71. R. W. Irving, An extension of Schur’s theorem on sum-free partitions, Acta Arith. 25 (1973) 55–63.MathSciNetMATHGoogle Scholar
  72. J. Komlbs, M. Sulyok and E. Szemerédi, Linear problems in combinatorial number theory, Acta Math. Acad. Sci. Hungar. 26 (1975) 113–121.MathSciNetCrossRefGoogle Scholar
  73. L. Mirsky, The combinatorics of arbitrary partitions, Bull. Inst. Math. Appl. 11 (1975) 6–9.MathSciNetGoogle Scholar
  74. I. Schur, Über die Kongruenz x`“ + y`” = z’“ (mod p), Jahresb. der Deutsche Math: Verein. 25 (1916) 114–117.MATHGoogle Scholar
  75. W. D. Wallis, A. P. Street, and J. S. Wallis, Combinatorics: Room Squares, Sum free Sets, Hadamard Matrices, Springer-Verlag, Heidelberg, 1972.Google Scholar
  76. S. Znâm, Generalisation of a number-theoretic result, Mat.-Fyz. Gasopis 16 (1966) 357–361.MATHGoogle Scholar
  77. S. Znâm, On k-thin sets and n-extensive graphs, Math. Gasopis 17 (1967) 297–307.MATHGoogle Scholar
  78. H. L. Abbott and E. T. H. Wang, Sum-free sets of integers, Proc. Amer. Math. Soc. 67 (1977) 11–16; MR 58 #5571.Google Scholar
  79. S. Znâm, Megjegyzések Turân Pal egy publikâlatlan ereményéhez, Mat. Lapok 14 (1963) 307–310.MathSciNetMATHGoogle Scholar
  80. Walter Deuber, Partitionen und lineare Gleichungssysteme, Math. Z. 133 (1973) 109–123.MathSciNetMATHCrossRefGoogle Scholar
  81. R. Rado, Studien zur Kombinatorik, Math. Z. 36 (1933) 424–480. E. R. Williams, M.Sc. thesis, Memorial Univ. 1967.Google Scholar
  82. Michael Beeler, William Gosper, and Rich Schroeppel, Hakmem, Memo 239, Artificial Intelligence Laboratory, M.I.T., 1972, p. 64.Google Scholar
  83. Corrado Böhm and Giovanna Sontacchi, On the existence of cycles of given length in integer sequences like x„+ 1 = x„/2 if x„ even, and x,„+ i = 3x„ + 1 otherwise, Atti Accad. Naz. Lincei Rend. Sci. Fis. Mat. Natur. (8) 64 (1978) 260–264.MATHGoogle Scholar
  84. R. E. Crandall, On the “3x + 1” problem, Math. Comput. 32 (1978)1281–1292; MR 58 #494; Zbl. 395. 10013.Google Scholar
  85. C. J. Everett, Iteration of the number-theoretic function f (2n) = n, f(2n + 1) = 3n + 2, Advances in Math., 25 (1977) 42–45; MR 56 #15552; Zbl. 352. 10001.Google Scholar
  86. Martin Gardner, Mathematical Games, A miscellany of transcendental problems, simple to state but not at all easy to solve, Scientific Amer. 226 #6 (Jun 1972) 114–118, esp p. 115.Google Scholar
  87. E. Heppner, Eine Bemerkung zum Hasse-Syracuse-Algorithmus, Arch. Math. (Basel) 31 (1977/79) 317–320; MR 80d:10007; Zbl. 377. 10027.Google Scholar
  88. David C. Kay, Pi Mu Epsilon J. 5 (1972) 338.Google Scholar
  89. H. Möller, Über Hasses Verallgemeinerung der Syracuse-Algorithmus (Kakutani’s Problem), Acta Arith. 34 (1978) 219–226; MR 57 #16246; Zbl. 329. 10008.Google Scholar
  90. Ray P. Steiner, A theorem on the Syracuse problem, Congressus Numerantium XX, Proc. 7th Conf. Numerical Math, Comput. Manitoba, 1977, 553–559; MR 80g: 10003.Google Scholar
  91. Riho Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976) 241–252; MR 58 #27879 (and see 35 (1979) 100–102; MR 80h: 10066.Google Scholar
  92. J. H. Conway, Unpredictable iterations, in Proc. Number Theory Conf., Boulder, 1972, 49–52.Google Scholar
  93. K. Mahler, An unsolved problem on the powers of 3/2, J. Austral. Math. Soc. 8 (1968) 313–321; MR 37 #2694.Google Scholar
  94. W. Forman and H. N. Shapiro, An arithmetic property of certain rational powers, Comm. Pure Appl. Math. 20 (1967) 561–573; MR 35 #2852.Google Scholar
  95. H. Davenport, A combinatorial problem connected with differential equations II (ed. A. Schinzel), Acta Arith. 17 (1971) 363–372.Google Scholar
  96. H. Davenport and A. Schinzel, A combinatorial problem connected with differential equations, Amer. J. Math. 87 (1965) 684–694.MathSciNetMATHCrossRefGoogle Scholar
  97. Annette J. Dobson and Shiela Oates Macdonald, Lower bounds for the lengths of Davenport—Schinzel sequences, Utilitas Math. 6 (1974) 251–257.MathSciNetMATHGoogle Scholar
  98. W. H. Mills, Some Davenport—Schinzel sequences, Congressus Numerantium IX,Proc. 3rd Manitoba Conf. Numerical Math. 1973, 307–313; MR 50 #135. Google Scholar
  99. C. R. Peterkin, Some results on Davenport—Schinzel sequences, Congressus Numerantium IX,Proc. 3rd Manitoba Conf. Numerical Math. 1973, 337–344; MR 50 #136.Google Scholar
  100. B. C. Rennie and A. J. Dobson, Upper bounds for the lengths of Davenport—Schinzel sequences, Utilitas Math. 8 (1975) 181–185.MathSciNetMATHGoogle Scholar
  101. D. P. Roselle, An algorithmic approach to Davenport—Schinzel sequences, Utilitas Math. 6 (1974) 91–93; MR 50 #9780.Google Scholar
  102. D. P. Roselle and R. G. Stanton, Results on Davenport—Schinzel sequences, Congressus Numerantium I, Proc. Louisiana Conf. Combin. Graph Theory, Comput. Baton Rouge, 1970, 249–267.Google Scholar
  103. R. G. Stanton and P. H. Dirksen, Davenport—Schinzel sequences, Ars Combinatoria 1 (1976) 43–51.MathSciNetMATHGoogle Scholar
  104. R. G. Stanton and R. C. Mullin, A map-theoretic approach to Davenport—Schinzel sequences, Pacific J. Math. 40 (1972) 167–172.MathSciNetMATHCrossRefGoogle Scholar
  105. R. G. Stanton and D. P. Roselle, A result on Davenport—Schinzel sequences, Colloq. Math. Soc. Janos Bolyai 4, Combinatorial Theory and its Applications, Balatonfüred, 1969, 1023–1027.Google Scholar
  106. R. G. Stanton and D. P. Roselle, Some properties of Davenport—Schinzel sequences, Acta Arith. 17 (1970–71) 355–362.Google Scholar
  107. S. Arshon, Démonstration de l’éxistence des suites asymétriques infinies (Russian. French summary), Mat. Sb. 2 (44) (1937) 769–779.Google Scholar
  108. C. H. Braunholtz, Solution to problem 5030 [1962, 439], Amer. Math. Monthly 70 (1963) 675–676.MathSciNetCrossRefGoogle Scholar
  109. T. C. Brown, Is there a sequence on four symbols in which no two adjacent segments are permutations of one another? Amer. Math. Monthly 78 (1971) 886–888.MathSciNetMATHCrossRefGoogle Scholar
  110. Richard A. Dean, A sequence without repeats on x, x-1, y, y-1, Amer. Math. Monthly 72 (1965) 383–385.MathSciNetCrossRefGoogle Scholar
  111. F. M. Dekking, On repetitions of blocks in binary sequences, J. Combin. Theory Ser. A, 20 (1976) 292–299.MathSciNetMATHCrossRefGoogle Scholar
  112. F. M. Dekking, Strongly non-repetitive sequences and progression-free sets, J. Combin. Theory Ser. A 27 (1979) 181–185.MathSciNetMATHCrossRefGoogle Scholar
  113. R. C. Entringer, D. E. Jackson and J. A. Schatz, On non-repetitive sequences, J. Comb in. Theory Ser. A, 16 (1974) 159–164.MathSciNetMATHCrossRefGoogle Scholar
  114. P. Erdös, Some unsolved problems, Magyar Tud. Akad. Mat. Kutat6 Int. Közl. 6 (1961) 221–254, esp. p. 240.Google Scholar
  115. A. A. Evdokimov, Strongly asymmetric sequences generated by a finite number of symbols, Dokl. Akad. Nauk SSSR 179 (1968) 1268–1271; Soviet Math. Dokl. 9 (1968) 536–539.MATHGoogle Scholar
  116. Earl Dennet Fife, Binary sequences which contain no BBb, PhD thesis, Wesleyan Univ., Middletown, Connecticut, 1976.Google Scholar
  117. D. Hawkins and W. E. Mientka, On sequences which contain no repetitions, Math. Student 24 (1956) 185–187; MR 19, 241.Google Scholar
  118. G. A. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk Mat. Tidskr. 15 (1967) 147–150; MR 37 #4454.Google Scholar
  119. G. A. Hedlund and W. H. Gottschalk, A characterization of the Morse minimal set, Proc. Amer. Math. Soc. 16 (1964) 70–74.MathSciNetGoogle Scholar
  120. J. Justin, Généralisation du théorème de van der Waerden sur les semi-groupes répétitifs, J. Combin. Theory Ser A, 12 (1972) 357–367.MathSciNetMATHCrossRefGoogle Scholar
  121. J. Justin, Semi-groupes répétitifs, Sém. IRIA, Log. Automat. 1971, 101–105, 108; Zbl. 274. 20092.Google Scholar
  122. J. Justin, Characterization of the repetitive commutative semigroups, J. Algebra 21 (1972) 87–90; MR 46 #277; Zbl. 248. 05004.Google Scholar
  123. John Leech, A problem on strings of beads, Math. Gaz. 41 (1957) 277–278.MATHCrossRefGoogle Scholar
  124. Marston Morse, A solution of the problem of infinite play in chess, Bull. Amer. Math. Soc. 44 (1938) 632.Google Scholar
  125. Marston Morse and Gustav A. Hedlund, Unending chess, symbolic dynamics and a problem in semigroups, Duke Math. J. 11 (1944) 1–7; MR 5 (1944) 202.Google Scholar
  126. P. A. B. Pleasants, Non-repetitive sequences, Proc. Cambridge Philos. Soc. 68 (1970) 267–274.MathSciNetMATHCrossRefGoogle Scholar
  127. Helmut Prodinger and Friedrich J. Urbanek, Infinite 0–1 sequences without long adjacent identical blocks, Discrete Math. 28 (1979) 277–289.MathSciNetMATHCrossRefGoogle Scholar
  128. H. E. Robbins, On a class of recurrent sequences, Bull. Amer. Math. Soc. 43 (1937) 413–417.MathSciNetCrossRefGoogle Scholar
  129. A. Thue, Über unendliche Zeichenreihen, Norse Vid. Selsk. Skr. I Mat.-Nat. Kl. Christiania (1906), No. 7, 1–22.Google Scholar
  130. A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, ibid. (1912), No. 1, 1–67.Google Scholar
  131. R. B. Crittenden and C. L. Vanden Eynden, Any n arithmetic progressions covering the first 2“ integers covers all integers, Proc. Amer. Math. Soc. 24 (1970) 475–481.MathSciNetMATHGoogle Scholar
  132. R. B. Crittenden and C. L. Vanden Eynden, The union of arithmetic progressions with differences not less than k, Amer. Math. Monthly 79 (1972) 630.MATHCrossRefGoogle Scholar
  133. P. Erdös, Some problems and results on the irrationality of the sum of infinite series, J. Math. Sci. 10 (1975) 1–7.MathSciNetCrossRefGoogle Scholar
  134. E. R. Berlekamp, J. H. Conway, and R. K. Guy, Winning Ways,Academic Press, London, 1981, Chap. 15.Google Scholar
  135. E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning Ways, Academic Press, London, 1981, Chapter 4.Google Scholar
  136. R. B. Eggleton, Generalized integers, M.A. Thesis, Univ. of Melbourne, 1969.Google Scholar
  137. Eugene Levine, An extremal result for sum-free sequences, J. Number Theory, 12 (1980) 251–257.MathSciNetMATHCrossRefGoogle Scholar
  138. Eugene Levine and Joseph O’Sullivan, An upper estimate for the reciprocal sum of a sum-free sequence, Acta Arith. 34 (1977) 9–24; MR 57 #5900; Zbl. 335. 10053.Google Scholar
  139. Abdul Majid Mian and S. D. Chowla, On the B2 sequences of Sidon, Proc. Nat. Acad. Sci. India Sect. A 14 (1944) 3–4; MR 7–243.Google Scholar
  140. J. O’Sullivan, On reciprocal sums of sum-free sequences, PhD thesis, Adelphi Univ. 1973.Google Scholar
  141. J. Baumgartner, A short proof of Hindman’s theorem, J. Combin. Theory Ser. A 17 (1974) 384–386.MathSciNetMATHCrossRefGoogle Scholar
  142. Neil Hindman, Finite sums with sequences within cells of a partition of n, J. Combin. Theory Ser. A 17 (1974) 1–11.MATHCrossRefGoogle Scholar
  143. Neil Hindman, Partitions and sums and products of integers, Trans. Amer. Math. Soc. 247 (1979) 227–245; MR 80b: 10022.Google Scholar
  144. Neil Hindman, Partitions and sums and products-two counterexamples, J. Combin. Theory Ser. A 29 (1980) 113–120.MathSciNetMATHCrossRefGoogle Scholar
  145. G. E. Andrews, MacMahon’s prime numbers of measurement, Amer. Math. Monthly 82 (1975) 922–923.MathSciNetMATHCrossRefGoogle Scholar
  146. R. L. Graham, Problem #1910, Amer. Math. Monthly 73 (1966) 775; solution, 75 (1968) 80–81.Google Scholar
  147. Jeff Lagarias, Problem 17, W. Coast Number Theory Conf., Asilomar, 1975.Google Scholar
  148. P. A. MacMahon, The prime numbers of measurement on a scale, Proc. Cambridge Philos. Soc. 21 (1923) 651–654.MATHGoogle Scholar
  149. Stefan Porubskÿ, On MacMahon’s segmented numbers and related sequences, Nieuw Arch. Wisk. (3) 25 (1977) 403–408; MR 58 #5575.Google Scholar
  150. N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, New York, 1973; sequences 363, 416, 1044.Google Scholar
  151. P. Erdös and R. L. Graham, Old and New Problems and Results in Combinatorial Number Theory, Monographie de L’Enseignement Mathématique No. 28, Genève, 1980, pp. 83–84.Google Scholar
  152. L. E. Dickson, The converse of Waring’s problem, Bull. Amer. Math. Soc. 40 (1934) 711–714.MathSciNetCrossRefGoogle Scholar
  153. J. A. Davis, R. C. Entringer, R. L. Graham and G. J. Simmons, On permutations containing no long arithmetic progressions, Acta Arith. 34 (1977) 81–90; MR 58 #10705.Google Scholar
  154. Tom Odda, Solution to Problem E2440, Amer. Math. Monthly 82 (1975) 74.MathSciNetCrossRefGoogle 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

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