A395458 a(n) = n*tau(n) - (n-1)*tau(rad(n)).

1, 2, 2, 6, 2, 4, 2, 18, 11, 4, 2, 28, 2, 4, 4, 50, 2, 40, 2, 44, 4, 4, 2, 100, 27, 4, 56, 60, 2, 8, 2, 130, 4, 4, 4, 184, 2, 4, 4, 164, 2, 8, 2, 92, 94, 4, 2, 292, 51, 104, 4, 108, 2, 220, 4, 228, 4, 4, 2, 248, 2, 4, 130, 322, 4, 8, 2, 140, 4, 8, 2, 580, 2, 4, 154, 156, 4, 8, 2, 484, 245, 4, 2, 344, 4, 4, 4, 356

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..88.

Formula

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

a(p^k) = p^k*(k-1)+2 for p prime and k>=1. - Wesley Ivan Hurt, May 11 2026

Example

a(12) = 12^0 + 12^1 + 12^0 + 12^0 + 12^1 + 12^0 = 28.

Mathematica

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

Crossrefs

Cf. A000005 (tau), A007947 (rad), A034444, A395435.

Sequence in context: A253139 A318519 A349356 * A317848 A124859 A021446

Adjacent sequences: A395455 A395456 A395457 * A395459 A395460 A395461

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 23 2026

Status

approved