A394747 Numbers k such that k-4 and k+4 are prime and gcd(k, sigma(k)) is prime.

15, 33, 105, 177, 195, 267, 273, 393, 405, 483, 573, 705, 747, 825, 915, 933, 987, 1017, 1065, 1113, 1167, 1233, 1293, 1323, 1455, 1527, 1563, 1605, 1617, 1827, 2157, 2247, 2337, 2343, 2463, 2535, 2553, 2703, 2715, 2745, 2883, 2913, 2967, 3015, 3255, 3327, 3453, 3495, 3537, 3543

Offset

1,1

Comments

k is the midpoint of a prime pair (k-4, k+4) differing by 8; this sequence is a subsequence of A087680.

All terms are divisible by 3. Since k-4 and k+4 are odd primes greater than 3, they must be congruent to 5 and 1 mod 6, respectively; hence all terms are congruent to 3 mod 6.

Most computed terms satisfy gcd(k, sigma(k)) = 3, though other prime values such as 7 and 13 also occur.

Links

Table of n, a(n) for n=1..50.

Example

For k = 15: 15-4 = 11 and 15+4 = 19 are prime. Also, sigma(15) = 24 and gcd(15, 24) = 3, which is prime, so 15 is a term.
For k = 9: 9-4 = 5 and 9+4 = 13 are prime, but sigma(9) = 13 and gcd(9, 13) = 1, which is not prime, so 9 is not a term.

Mathematica

Select[Range[3600], And @@ PrimeQ[{# - 4, # + 4, GCD[#, DivisorSigma[1, #]]}] &] (* Amiram Eldar, May 08 2026 *)

Prog

(Python)
from sympy import divisor_sigma, gcd, isprime
def ok(k): return isprime(k-4) and isprime(k+4) and isprime(gcd(k, divisor_sigma(k)))
print([k for k in range(1, 4000) if ok(k)])

Crossrefs

Intersection of A087680 and A392199.

Cf. A394757, A395258.

Sequence in context: A211327 A222179 A322493 * A190052 A242243 A219855

Adjacent sequences: A394744 A394745 A394746 * A394748 A394749 A394750

Keyword

nonn

Author

Aied Sulaiman, May 08 2026

Status

approved