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

Author

Wesley Ivan Hurt, Jun 21 2025

Keywords

• nonn
• easy

Comments

Inverse Möbius transform of A103164(n).

Examples

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

Crossrefs

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

Programs

Mathematica

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

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