A365682 The sum of exponentially squarefree divisors of n.

1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, 14, 24, 24, 15, 18, 39, 20, 42, 32, 36, 24, 60, 31, 42, 40, 56, 30, 72, 32, 47, 48, 54, 48, 91, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 60, 57, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 111, 84, 144

Offset

1,2

Comments

The sum of divisors of n that are exponentially squarefree numbers (A209061), i.e., numbers having only squarefree exponents in their canonical prime factorization.

The number of these divisors is A365680(n) and the largest of them is A365683(n).

Links

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

Formula

Multiplicative with a(p^e) = 1 + Sum_{k=1..e, k squarefree} p^k.

a(n) <= A000203(n), with equality if and only if n is a biquadratefree number (A046100).

Mathematica

f[p_, e_] := 1 + Sum[If[SquareFreeQ[k], p^k, 0], {k, 1, e}]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

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

Crossrefs

Cf. A000203, A046100, A209061, A365680, A365683.

Sequence in context: A284587 A097011 A366992 * A074847 A365170 A365174

Adjacent sequences: A365679 A365680 A365681 * A365683 A365684 A365685

Keyword

nonn,easy,mult

Author

Amiram Eldar, Sep 15 2023

Status

approved