A394757 - OEIS

%I #17 Apr 13 2026 18:21:14

%S 18,72,1152,4050,5652,6948,15138,16452,20772,20808,27792,44532,46818,

%T 65700,67428,126612,143568,176418,180288,194868,195732,206082,212868,

%U 216900,218628,236772,276372,281250,299700,328788,362952,369828,377172,388962,390672,400068,402768

%N Numbers k in A014574 (averages of twin prime pairs) such that gcd(k, sigma(k)) is prime.

%C Conjecture: all terms are divisible by 18.

%C Conjecture: gcd(k, sigma(k)) is in {2,3}.

%e For k = 18: sigma(18) = 39 and gcd(18, 39) = 3, which is prime, so 18 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[Array[Prime[#] + 1 &, 35000], PrimeQ[# + 1] && PrimeQ[GCD[#, DivisorSigma[1, #]]] &] (* _Amiram Eldar_, Mar 31 2026 *)

%o (Python)

%o from sympy import divisor_sigma, gcd, isprime

%o def ok(k): return isprime(k-1) and isprime(k+1) and isprime(gcd(k, divisor_sigma(k)))

%o print([k for k in range(1, 500000) if ok(k)])

%Y Intersection of A014574 and A392199.

%K nonn

%O 1,1

%A _Aied Sulaiman_, Mar 31 2026