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

Author

Amiram Eldar, Sep 15 2023

Keywords

• nonn
• easy
• mult

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

Crossrefs

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

Programs

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]

PARI

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

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).