A394569 Numbers k such that gcd(k, Omega(k)) is prime.

4, 6, 10, 12, 14, 18, 22, 26, 27, 30, 34, 38, 42, 45, 46, 54, 58, 62, 63, 64, 66, 74, 75, 78, 80, 82, 86, 90, 94, 99, 102, 105, 106, 114, 117, 118, 120, 122, 126, 134, 138, 142, 146, 147, 150, 153, 158, 160, 165, 166, 171, 174, 178, 180, 186, 194, 195, 198, 200, 202

Offset

1,1

Comments

No prime belongs to this sequence since, for any prime k, Omega(k) = 1 and gcd(k, 1) = 1, which is not prime.

2^n is a term if and only if n == 2 (mod 4), since Omega(2^n) = n and gcd(2^n, n) = 2 only in this case.

Links

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

Example

For k = 6: Omega(6) = 2 and gcd(6, 2) = 2, which is prime, so 6 is a term.
For k = 8: Omega(8) = 3 and gcd(8, 3) = 1, which is not prime, so 8 is not a term.

Mathematica

Select[Range[200], PrimeQ[GCD[#, PrimeOmega[#]]] &] (* Amiram Eldar, Mar 25 2026 *)

Prog

(Python)
from sympy import factorint, gcd, isprime
def ok(k): return isprime(gcd(k, sum(factorint(k).values())))
print([k for k in range(1, 400) if ok(k)])

Crossrefs

Cf. A001222, A392199, A394433.

Sequence in context: A181794 A199536 A284883 * A134333 A114331 A102070

Adjacent sequences: A394566 A394567 A394568 * A394570 A394571 A394572

Keyword

nonn,easy

Author

Aied Sulaiman, Mar 25 2026

Status

approved