A347038 - OEIS

A347038

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

29, 37, 41, 53, 67, 89, 101, 109, 113, 127, 137, 151, 157, 173, 181, 197, 227, 229, 233, 257, 269, 277, 281, 293, 313, 349, 373, 389, 401, 409, 421, 439, 461, 557, 587, 593, 601, 613, 617, 641, 643, 653, 661, 673, 677, 701, 709, 739, 761, 773, 787, 821, 829

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

MAPLE

filter:= proc(p) isprime(p) and not ormap(k -> numtheory:-tau(k+p) = numtheory:-sigma(k), [$1 .. 1 + 2*isqrt(p)]) end proc:

select(filter, [seq(i, i=3..1000, 2)]); # Robert Israel, Aug 06 2025

PROG

(Python)

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

from math import isqrt

def A347038_list(pmax):

a = []

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