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Primes p such that there are no solutions to d(k+p) = sigma(k).
0

%I #29 Sep 14 2021 04:25:46

%S 29,37,41,53,67,89,101,109,113,127,137,151,157,173,181,197,227,229,

%T 233,257,269,277,281,293,313,349,373,389,401,409,421,439,461,557,587,

%U 593,601,613,617,641,643,653,661,673,677,701,709,739,761,773,787,821,829

%N Primes p such that there are no solutions to d(k+p) = sigma(k).

%C If p is not in the sequence and d(k+p) = sigma(k), then k <= 1+2*sqrt(p). Proof: We have d(m) <= 2*sqrt(m) (see formula in A000005), so 2*sqrt(k+p) >= d(k+p) = sigma(k) >= k+1 (if k > 1). After squaring and simplifying, we get k <= 1+2*sqrt(p). - _Pontus von Brömssen_, Aug 20 2021

%o (Python)

%o from sympy import divisor_count as d, divisor_sigma as sigma, primerange

%o from math import isqrt

%o def A347038_list(pmax):

%o a = []

%o for p in primerange(2, pmax + 1):

%o if not any(d(k + p) == sigma(k) for k in range(1, 2 + isqrt(4 * p))):

%o a.append(p)

%o return a # Pontus von Brömssen, Aug 20 2021

%Y Cf. A000005, A000203, A247485.

%K nonn

%O 1,1

%A _Angad Singh_, Aug 12 2021