A392199 - OEIS

%I #24 Jan 09 2026 13:54:18

%S 10,14,15,18,20,22,26,33,34,38,45,46,51,52,58,62,68,69,70,72,74,80,82,

%T 86,87,91,94,95,99,104,105,106,110,116,117,118,122,123,130,134,136,

%U 141,142,145,146,147,148,154,158,159,160,162,164,165,166,170,177,178,194,195

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

%C 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.

%C 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

%C 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

%e For k = 10: sigma(10) = 18 and gcd(10, 18) = 2, which is prime, so 10 is a term.

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

%t Select[Range[200], PrimeQ[GCD[#, DivisorSigma[1, #]]] &] (* _Amiram Eldar_, Jan 03 2026 *)

%o (Python)

%o from sympy import divisor_sigma, gcd, isprime

%o def ok(k): return isprime(gcd(k, divisor_sigma(k)))

%o print([k for k in range(1, 200) if ok(k)])

%o (PARI) isok(k) = isprime(gcd(k, sigma(k))); \\ _Michel Marcus_, Jan 03 2026

%Y Cf. A009194, A014567, A205523.

%Y Subsequence of A069059.

%K nonn,easy

%O 1,1

%A _Aied Sulaiman_, Jan 03 2026