A386471 The sum of the divisors of n whose exponents in their prime factorization are squares.

1, 3, 4, 3, 6, 12, 8, 3, 4, 18, 12, 12, 14, 24, 24, 19, 18, 12, 20, 18, 32, 36, 24, 12, 6, 42, 4, 24, 30, 72, 32, 19, 48, 54, 48, 12, 38, 60, 56, 18, 42, 96, 44, 36, 24, 72, 48, 76, 8, 18, 72, 42, 54, 12, 72, 24, 80, 90, 60, 72, 62, 96, 32, 19, 84, 144, 68, 54

Offset

1,2

Comments

The sum of the terms in A197680 that divide n.

The number of these divisors is A386470(n) and the largest of them is A386469(n).

Links

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

Formula

Multiplicative with a(p^e) = Sum_{k=0..floor(sqrt(e))} p^(k^2).

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + Sum_{k>=2} Sum_{i=(2*k)^2..(2*k+1)^2-1} (-1/p)^i) = 1.05459216969289486594... .

Mathematica

f[p_, e_] := Sum[p^(k^2), {k, 0, Floor[Sqrt[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~, sum(k = 0, sqrtint(f[i, 2]), f[i, 1]^(k^2))); }

Crossrefs

Cf. A000203, A005117, A197680, A386469, A386470.

Sequence in context: A390666 A392167 A390664 * A073181 A183100 A340323

Adjacent sequences: A386468 A386469 A386470 * A386472 A386473 A386474

Keyword

nonn,easy,mult

Author

Amiram Eldar, Jul 23 2025

Status

approved