On primes and practical numbers

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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Acknowledgements

We thank David Eppstein for informing us of [7] and Paul Pollack for [19].

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Correspondence to Andreas Weingartner.

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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 (2021). https://doi.org/10.1007/s11139-020-00354-y

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Keywords

  • Practical number
  • Shifted prime

Mathematics Subject Classification

  • 11N25 (11N37)