A395301 a(n) = tau(n) - omega(n) + Sum_{p|n, p prime} gcd(p,n/p).

1, 2, 2, 4, 2, 4, 2, 5, 5, 4, 2, 7, 2, 4, 4, 6, 2, 8, 2, 7, 4, 4, 2, 9, 7, 4, 6, 7, 2, 8, 2, 7, 4, 4, 4, 12, 2, 4, 4, 9, 2, 8, 2, 7, 8, 4, 2, 11, 9, 10, 4, 7, 2, 10, 4, 9, 4, 4, 2, 13, 2, 4, 8, 8, 4, 8, 2, 7, 4, 8, 2, 15, 2, 4, 10, 7, 4, 8, 2, 11, 7, 4, 2, 13, 4, 4, 4, 9, 2, 14, 4, 7, 4, 4, 4, 13, 2, 12, 8, 14

Offset

1,2

Comments

For each divisor d of n, add gcd(d,n/d) if d is prime, else add 1.

Links

Table of n, a(n) for n=1..100.

Formula

a(n) = Sum_{d|n} gcd(d,n/d)^c(d), where c is the prime characteristic (A010051).

a(n) = A033273(n) + A345266(n).

a(p^k) = k+p^(1-floor(1/k)) for p prime and k >= 1. - Wesley Ivan Hurt, May 23 2026

Mathematica

Table[Sum[GCD[d, n/d]^(PrimePi[d] - PrimePi[d - 1]), {d, Divisors[n]}], {n, 100}]

Crossrefs

Cf. A000005 (tau), A001221 (omega), A010051, A033273, A345266.

Sequence in context: A249867 A351612 A092520 * A372714 A147848 A306652

Adjacent sequences: A395298 A395299 A395300 * A395302 A395303 A395304

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 18 2026

Status

approved