A360159 a(n) is the sum of divisors of n that are odd squares.

1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 26, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 50, 26, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 1, 10, 1, 1, 26, 1, 1, 1, 1, 1, 91, 1, 1

Offset: 1

Author

Amiram Eldar, Jan 29 2023

Keywords

• nonn
• easy
• mult

Links

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

Crossrefs

Cf. A016754, A035316, A078434, A344300.

Programs

Mathematica

f[p_, e_] := (p^(2*(1 + Floor[e/2])) - 1)/(p^2 - 1); f[2, e_] := 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, 1] == 2, 1, (f[i, 1]^(2*(f[i, 2]\2)+2)-1)/(f[i, 1]^2-1))); }

Formula

a(n) = Sum_{d|n, d odd square} d.

a(n) = (A035316(n) + A344300(n))/2.

Multiplicative with a(2^e) = 1, and for p > 2, a(p^e) = (p^(e+2)-1)/(p^2-1) for even e and a(p^e) = (p^(e+1)-1)/(p^2-1) for odd e.

Dirichlet g.f.: zeta(s)*zeta(2s-2)*(1-4^(1-s)).

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)/6 = 0.4353958914... .