A394651 a(n) = tau(n) + Sum_{p|n, p prime, gcd(p,n/p) = 1} (p-1).

1, 3, 4, 3, 6, 7, 8, 4, 3, 9, 12, 8, 14, 11, 10, 5, 18, 7, 20, 10, 12, 15, 24, 10, 3, 17, 4, 12, 30, 15, 32, 6, 16, 21, 14, 9, 38, 23, 18, 12, 42, 17, 44, 16, 10, 27, 48, 12, 3, 7, 22, 18, 54, 9, 18, 14, 24, 33, 60, 18, 62, 35, 12, 7, 20, 21, 68, 22, 28, 19, 72, 12, 74, 41, 8, 24, 20, 23, 80, 14, 5, 45, 84, 20, 24

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..85.

Formula

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

a(n) = A000005(n) + A063956(n) - A056169(n).

a(p^k) = k + p^floor(1/k) for p prime and k >= 1. - Wesley Ivan Hurt, May 15 2026

Example

a(30) = 1^0 + 2^1 + 3^1 + 5^1 + 6^0 + 10^0 + 15^0 + 30^0 = 15.

Mathematica

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

Prog

(PARI) a(n) = my(f=factor(n)); numdiv(f) + sum(k=1, #f~, my(p=f[k, 1]); if (gcd(p, n/p) == 1, p-1)); \\ Michel Marcus, Apr 25 2026

Crossrefs

Cf. A000005 (tau), A010051, A056169, A063956.

Sequence in context: A049276 A101684 A061800 * A382715 A384125 A218789

Adjacent sequences: A394648 A394649 A394650 * A394652 A394654 A394655

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 24 2026

Status

approved