A395491 a(n) = tau(n) + (n-1) * Sum_{p|n, p prime} [gcd(p,n/p) = 1], where [ ] is the Iverson bracket.

1, 3, 4, 3, 6, 14, 8, 4, 3, 22, 12, 17, 14, 30, 32, 5, 18, 23, 20, 25, 44, 46, 24, 31, 3, 54, 4, 33, 30, 95, 32, 6, 68, 70, 72, 9, 38, 78, 80, 47, 42, 131, 44, 49, 50, 94, 48, 57, 3, 55, 104, 57, 54, 61, 112, 63, 116, 118, 60, 130, 62, 126, 68, 7, 132, 203, 68, 73, 140, 215, 72, 12, 74, 150, 80, 81, 156, 239, 80, 89, 5, 166

Offset

1,2

Comments

For each divisor d of n, add n if d is a unitary prime divisor of n, else add 1.

Links

Table of n, a(n) for n=1..82.

Formula

a(n) = Sum_{d|n} n^([gcd(d,n/d) = 1] * c(d)), where [ ] is the Iverson bracket and c = A010051.

a(n) = A000005(n) + (n-1) * A056169(n).

a(p^k) = k+1+floor(1/k)*(p-1) for p prime and k>=1. - Wesley Ivan Hurt, May 11 2026

Example

a(24) = 24^0 + 24^0 + 24^1 + 24^0 + 24^0 + 24^0 + 24^0 + 24^0 = 31.

Mathematica

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

Crossrefs

Cf. A000005 (tau), A010051, A056169, A394651.

Sequence in context: A366764 A365337 A349337 * A377604 A358045 A371858

Adjacent sequences: A395488 A395489 A395490 * A395492 A395493 A395494

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 25 2026

Status

approved