A343442 If n = Product (p_j^k_j) then a(n) = Product (p_j + 2), with a(1) = 1.

1, 4, 5, 4, 7, 20, 9, 4, 5, 28, 13, 20, 15, 36, 35, 4, 19, 20, 21, 28, 45, 52, 25, 20, 7, 60, 5, 36, 31, 140, 33, 4, 65, 76, 63, 20, 39, 84, 75, 28, 43, 180, 45, 52, 35, 100, 49, 20, 9, 28, 95, 60, 55, 20, 91, 36, 105, 124, 61, 140, 63, 132, 45, 4, 105, 260, 69, 76, 125, 252

Offset: 1

Author

Ilya Gutkovskiy, Apr 15 2021

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from indices in prime factorization.

Crossrefs

Cf. A000203, A001615, A007429, A007947, A008683, A048250, A062953, A063441, A074816, A092261, A107758, A107759, A319132, A335005.

Programs

Mathematica

a[1] = 1; a[n_] := Times @@ ((#[[1]] + 2) & /@ FactorInteger[n]); Table[a[n], {n, 70}]
nmax = 70; CoefficientList[Series[Sum[MoebiusMu[k]^2 DivisorSigma[1, k] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

PARI

a(n) = sumdiv(n, d, moebius(d)^2 * sigma(d)) \\ Andrew Howroyd, Apr 15 2021

Formula

G.f.: Sum_{k>=1} mu(k)^2 * sigma(k) * x^k / (1 - x^k), where mu = A008683 and sigma = A000203.

a(n) = Sum_{d|n} mu(d)^2 * sigma(d).

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/(12*zeta(3)) = 0.684216... (A335005). - Amiram Eldar, Nov 13 2022

a(n) = Sum_{d|n} mu(d)^2*psi(d), where psi is A001615. - Ridouane Oudra, Jul 24 2025