A380444 Sum of the nonprimes dividing n and the squares of the primes dividing n.

1, 5, 10, 9, 26, 20, 50, 17, 19, 40, 122, 36, 170, 68, 50, 33, 290, 47, 362, 64, 80, 148, 530, 68, 51, 200, 46, 100, 842, 100, 962, 65, 164, 328, 110, 99, 1370, 404, 218, 112, 1682, 146, 1850, 196, 104, 580, 2210, 132, 99, 115, 350, 256, 2810, 128, 202, 164, 428, 904, 3482, 196, 3722, 1028, 152, 129, 260, 262, 4490, 400, 608, 208, 5042, 203, 5330, 1448, 150, 484

Offset

1,2

Comments

Inverse Möbius transform of A103164(n).

Links

Table of n, a(n) for n=1..76.

Formula

a(n) = sigma(n) - sopf(n) + sopf_2(n), where sopf_2(n) = Sum_{p|n, p prime} p^2.

a(n) = Sum_{d|n} d^tau(d^c(d)), where c = A010051.

a(n) = A023890(n) + A005063(n).

a(p^k) = (p^(k+1)+p^3-2*p^2+p-1)/(p-1) for p prime, k >= 1. - Wesley Ivan Hurt, Jul 02 2025

Example

a(12) = 1 + 2^2 + 3^2 + 4 + 6 + 12 = 36.

Mathematica

Table[DivisorSigma[1, n] + Sum[p (p - 1), {p, Select[Divisors[n], PrimeQ]}], {n, 100}]

Crossrefs

Cf. A000005 (tau), A000203 (sigma), A005063, A008472 (sopf), A010051, A023890, A103164.

Sequence in context: A141622 A144136 A343852 * A198286 A001483 A173679

Adjacent sequences: A380441 A380442 A380443 * A380445 A380446 A380447

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Jun 21 2025

Status

approved