A367991 The sum of the divisors of the squarefree part of n.

1, 3, 4, 1, 6, 12, 8, 3, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 12, 1, 42, 4, 8, 30, 72, 32, 3, 48, 54, 48, 1, 38, 60, 56, 18, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 12, 72, 24, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72

Offset: 1

Author

Amiram Eldar, Dec 07 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A348503 at n = 72 and from A344695 at n = 108.

The sum of the infinitary divisors (A077609) of n that are squarefree (A005117). - Amiram Eldar, Jun 03 2025

Links

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

Crossrefs

Cf. A000203, A000290, A005117, A007913, A048250, A077609, A367990.

Cf. A002117, A013662.

Cf. A344695, A348503.

Programs

Mathematica

f[p_, e_] := If[OddQ[e], p + 1, 1]; 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]%2, f[i,1]+1, 1));}

Formula

Multiplicative with a(p^e) = p + 1 if e is odd and 1 otherwise.

a(n) = A000203(A007913(n)) = A048250(A007913(n)).

a(n) = A048250(n)/A367990(n).

a(n) >= 1, with equality if and only if n is a square (A000290).

a(n) <= A000203(n), with equality if and only if n is squarefree (A005117).

Dirichlet g.f.: zeta(2*s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(4)/zeta(3) = 0.900392677639... .