OFFSET
1,2
COMMENTS
The Mobius transform of this sequence appears to generate the sequence of absolute terms of A061020. - R. J. Mathar, Feb 08 2011
LINKS
Ivan Neretin, Table of n, a(n) for n = 1..10000
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
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
mult,nonn
AUTHOR
Vladeta Jovovic, Nov 12 2002
STATUS
approved