A387633 a(n) = tau(n) + (n-1) * tau(rad(n)).

1, 4, 6, 9, 10, 24, 14, 18, 19, 40, 22, 50, 26, 56, 60, 35, 34, 74, 38, 82, 84, 88, 46, 100, 51, 104, 56, 114, 58, 240, 62, 68, 132, 136, 140, 149, 74, 152, 156, 164, 82, 336, 86, 178, 182, 184, 94, 198, 99, 202, 204, 210, 106, 220, 220, 228, 228, 232, 118, 484, 122, 248, 254, 133, 260, 528, 134, 274, 276, 560, 142

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..71.

Formula

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

From Wesley Ivan Hurt, May 23 2026: (Start)

a(n) = A000005(n) + (n-1)*A034444(n).

a(p^k) = 2*p^k+k-1 for p prime and k>=0. (End)

Example

a(24) = 24^1 + 24^0 + 24^1 + 24^0 + 24^0 + 24^1 + 24^0 + 24^1 = 100.

Mathematica

Table[DivisorSigma[0, n] + (n - 1) DivisorSigma[0, Product[d^(PrimePi[d] - PrimePi[d - 1]), {d, Divisors[n]}]], {n, 100}]
(* Alternative: *)
Table[Sum[n^KroneckerDelta[GCD[d, n/d], 1], {d, Divisors[n]}], {n, 100}]

Crossrefs

Cf. A000005 (tau), A007947 (rad), A034444, A394481.

Sequence in context: A085842 A356135 A131220 * A332618 A295329 A302990

Adjacent sequences: A387630 A387631 A387632 * A387634 A387635 A387636

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Apr 20 2026

Status

approved