A065745 Sum of squares and twice squares dividing n.

1, 3, 1, 7, 1, 3, 1, 15, 10, 3, 1, 7, 1, 3, 1, 31, 1, 30, 1, 7, 1, 3, 1, 15, 26, 3, 10, 7, 1, 3, 1, 63, 1, 3, 1, 70, 1, 3, 1, 15, 1, 3, 1, 7, 10, 3, 1, 31, 50, 78, 1, 7, 1, 30, 1, 15, 1, 3, 1, 7, 1, 3, 10, 127, 1, 3, 1, 7, 1, 3, 1, 150, 1, 3, 26, 7, 1, 3, 1, 31, 91, 3, 1, 7, 1, 3, 1, 15, 1, 30, 1

Offset

1,2

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

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

From Amiram Eldar, Dec 15 2023: (Start)

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

Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = ((2+sqrt(2))/6) * zeta(3/2) = 1.4865345575818562471... . (End)

Mathematica

f[2, e_] := 2^(e+1) - 1; f[p_, e_] := If[OddQ[e], (p^(e+1)-1)/(p^2-1), (p^(e+2)-1)/(p^2-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 13 2020 *)

Prog

(PARI) a(n) = sumdiv(n, d, issquare(d)*d + (1 - d%2)*issquare(d/2)*d) \\ Michel Marcus, Jun 17 2013

Crossrefs

Cf. A035316, A065704, A078434.

Sequence in context: A053381 A038712 A354587 * A361438 A268670 A360331

Adjacent sequences: A065742 A065743 A065744 * A065746 A065747 A065748

Keyword

mult,nonn,easy

Author

Vladeta Jovovic, Dec 04 2001

Extensions

More terms from Matthew Conroy, Jan 19 2002

Status

approved