A076752 a(n) = Sum_{d is a square divisor of n} n/d.

1, 2, 3, 5, 5, 6, 7, 10, 10, 10, 11, 15, 13, 14, 15, 21, 17, 20, 19, 25, 21, 22, 23, 30, 26, 26, 30, 35, 29, 30, 31, 42, 33, 34, 35, 50, 37, 38, 39, 50, 41, 42, 43, 55, 50, 46, 47, 63, 50, 52, 51, 65, 53, 60, 55, 70, 57, 58, 59, 75, 61, 62, 70, 85, 65, 66, 67, 85, 69, 70, 71

Offset: 1

Author

Vladeta Jovovic, Nov 12 2002

Keywords

• mult
• nonn

Comments

The Mobius transform of this sequence appears to generate the sequence of absolute terms of A061020. - R. J. Mathar, Feb 08 2011

Examples

			a(8) = 10 as the square divisors of 8 are 1 and 4, and 8/1 + 8/4 = 10. - _David A. Corneth_, Nov 03 2017
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A008836, A035316.

Programs

Mathematica

Table[Total[n/Select[Divisors[n], IntegerQ@Sqrt@# &]], {n, 71}] (* Ivan Neretin, Sep 20 2017 *)
Table[DivisorSum[n, n/# &, IntegerQ@ Sqrt@ # &], {n, 71}] (* Michael De Vlieger, Nov 03 2017 *)
f[p_, e_] := p^(k = If[EvenQ[e], 0, 1])*(p^(e + 2 - k) - 1)/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

PARI

a(n,f=factor(n))=prod(i=1,#f~, if(f[i,2]%2, f[i,1]*(f[i,1]^(f[i,2]+1)-1), (f[i,1]^(f[i,2]+2)-1))/(f[i,1]^2-1)) \\ Charles R Greathouse IV, Sep 20 2017

PARI

a(n) = sumdiv(n, d, (n/d)*issquare(d)); \\ Michel Marcus, Nov 02 2017

Formula

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

a(p ^ (m + 1)) = p * a(p^m) for even m and a(p ^ (m + 1)) = p * a(p^m) + 1 for odd m. - David A. Corneth, Nov 03 2017

a(n) = (lambda * sigma)(n) = (A008836 * A000203)(n), where * is the Dirichlet convolution. - Yuyang Zhao, Nov 02 2017

From Vaclav Kotesovec, Feb 04 2019: (Start)

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

Sum_{k=1..n} a(k) ~ Pi^4 * n^2 / 180. (End)

G.f.: Sum_{k>=1} x^(k^2) / (1 - x^(k^2))^2. - Ilya Gutkovskiy, Aug 19 2021