2020/07/21 by Pomerance, Carl, Weingartner, Andreas
#11N25 (11N37) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2007.11062
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.