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

Author

Amiram Eldar, Jul 23 2025

Keywords

• nonn
• easy
• mult

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

Crossrefs

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

Programs

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]

PARI

a(n) = {my(f = factor(n)); prod(i = 1, #f~, sum(k = 0, sqrtint(f[i,2]), f[i,1]^(k^2)));}

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... .