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

1,4

Comments

The number of these divisors is A323308(n).

Links

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

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... .

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A222119 A351086 A102280 * A365403 A373439 A035316

Adjacent sequences: A370236 A370237 A370238 * A370240 A370241 A370242

Keyword

nonn,easy,mult

Author

Amiram Eldar, Feb 13 2024

Status

approved