Abstract
A number n is practical if every integer in [1, n] can be expressed as a subset sum of the positive divisors of n. We consider the distribution of practical numbers that are also shifted primes, improving a theorem of Guo and Weingartner. In addition, essentially proving a conjecture of Margenstern, we show that all large odd numbers are the sum of a prime and a practical number. We also consider an analogue of the prime k-tuples conjecture for practical numbers, proving the “correct” upper bound, and for pairs, improving on a lower bound of Melfi.
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In memory of Ron Graham (1935–2020) and Richard Guy (1916–2020)
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Pomerance, C., Weingartner, A. On primes and practical numbers. Ramanujan J 57, 981–1000 (2022). https://doi.org/10.1007/s11139-020-00354-y
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DOI: https://doi.org/10.1007/s11139-020-00354-y