A392199 Positive integers k such that gcd(k, sigma(k)) is prime.

10, 14, 15, 18, 20, 22, 26, 33, 34, 38, 45, 46, 51, 52, 58, 62, 68, 69, 70, 72, 74, 80, 82, 86, 87, 91, 94, 95, 99, 104, 105, 106, 110, 116, 117, 118, 122, 123, 130, 134, 136, 141, 142, 145, 146, 147, 148, 154, 158, 159, 160, 162, 164, 165, 166, 170, 177, 178, 194, 195

Offset

1,1

Comments

No prime k belongs to the sequence, since a prime has divisors 1 and k, so sigma(k) = 1 + k and gcd(k, sigma(k)) = 1, which is not prime.

If p > 3 is a prime, then gcd(2*p, sigma(2*p)) = gcd(2*p, 3*(p+1)) = 2, so 2*p is a term. - Amiram Eldar, Jan 03 2026

If p and q are primes (other than 2 and 3) with p | q + 1, then p*q is a term. - Robert Israel, Jan 04 2026

Links

Table of n, a(n) for n=1..60.

Example

For k = 10: sigma(10) = 18 and gcd(10, 18) = 2, which is prime, so 10 is a term.
For k = 12: sigma(12) = 28 and gcd(12, 28) = 4, which is not prime, so 12 is not a term.

Mathematica

Select[Range[200], PrimeQ[GCD[#, DivisorSigma[1, #]]] &] (* Amiram Eldar, Jan 03 2026 *)

Prog

(Python)
from sympy import divisor_sigma, gcd, isprime
def ok(k): return isprime(gcd(k, divisor_sigma(k)))
print([k for k in range(1, 200) if ok(k)])
(PARI) isok(k) = isprime(gcd(k, sigma(k))); \\ Michel Marcus, Jan 03 2026

Crossrefs

Cf. A009194, A014567, A205523.

Subsequence of A069059.

Sequence in context: A280032 A227010 A246449 * A121836 A317590 A081062

Adjacent sequences: A392196 A392197 A392198 * A392200 A392201 A392202

Keyword

nonn,easy

Author

Aied Sulaiman, Jan 03 2026

Status

approved