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

1,2

Comments

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

Links

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

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... .

Mathematica

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

Crossrefs

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

Cf. A002117, A013662.

Cf. A344695, A348503.

Sequence in context: A366795 A092261 A380087 * A344695 A348503 A348047

Adjacent sequences: A367988 A367989 A367990 * A367992 A367993 A367994

Keyword

nonn,easy,mult

Author

Amiram Eldar, Dec 07 2023

Status

approved