A285895 Sum of divisors d of n such that n/d is not congruent to 0 mod 4.

1, 3, 4, 6, 6, 12, 8, 12, 13, 18, 12, 24, 14, 24, 24, 24, 18, 39, 20, 36, 32, 36, 24, 48, 31, 42, 40, 48, 30, 72, 32, 48, 48, 54, 48, 78, 38, 60, 56, 72, 42, 96, 44, 72, 78, 72, 48, 96, 57, 93, 72, 84, 54, 120, 72, 96, 80, 90, 60, 144, 62, 96, 104, 96, 84, 144, 68

Offset: 1

Author

Seiichi Manyama, Apr 28 2017

Keywords

• nonn
• mult

Examples

			The divisors of 8 are 1, 2, 4, and 8. 8/1 == 0 (mod 4) and 8/2 == 0 (mod 4). Hence, a(8) = 4 + 8 = 12.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002131 (k=2), A078708 (k=3), this sequence (k=4), A285896 (k=5).

Cf. A002131, A050460, A050464.

Programs

Mathematica

f[p_, e_] := If[p == 2, 3*2^(e-1), (p^(e+1)-1)/(p-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)

PARI

a(n)=sumdiv(n, d, if(n/d%4, d, 0)); \\ Andrew Howroyd, Jul 20 2018

Formula

G.f.: Sum_{k>=1} k*x^k*(1 + x^k + x^(2*k))/(1 - x^(4*k)). - Ilya Gutkovskiy, Sep 12 2019

a(n) = A050460(n) + A002131(n/2) + A050464(n), where A002131(.)=0 for non-integer argument. - R. J. Mathar, May 25 2020

From Amiram Eldar, Oct 30 2022: (Start)

Multiplicative with a(2^e) = 3*2^(e-1) and a(p^e) = (p^(e+1)-1)/(p-1) if p > 2.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 5*Pi^2/64 = 0.7710628... . (End)

Dirichlet g.f.: zeta(s)*zeta(s-1)*(1-1/4^s). - Amiram Eldar, Dec 30 2022