A370239 The sum of divisors of n that are squares of squarefree numbers.

1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 1, 1, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 26, 1, 10, 5, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 1, 5, 1, 1, 1, 5, 10, 1, 1, 5, 50, 26, 1, 5, 1, 10, 1, 5, 1, 1, 1, 5, 1, 1, 10, 5, 1, 1, 1, 5, 1, 1, 1, 50, 1, 1, 26, 5, 1, 1, 1, 5, 10, 1, 1

Offset: 1

Author

Amiram Eldar, Feb 13 2024

Keywords

• nonn
• easy
• mult

Comments

The number of these divisors is A323308(n).

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A005117, A036785, A048250, A062503, A071327, A078434, A323308, A370240.

Programs

Mathematica

f[p_, e_] := If[e == 1, 1, 1 + p^2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,2] == 1, 1, 1 + f[i,1]^2));}

Formula

Multiplicative with a(p) = 1 and a(p^e) = 1 + p^2 for e >= 2.

a(n) >= 1, with equality if and only if n is squarefree (A005117).

a(n) = A071327(n) + 1 if and only if n is not in A036785.

Dirichlet g.f.: zeta(s)*zeta(2*s-2)/zeta(4*s-4).

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = 2*zeta(3/2)/Pi^2 = 0.5293779248... .