OFFSET
1,2
COMMENTS
More generally, a(n,m) = Sum_{d divides n} gcd(d,n/d)^m is multiplicative with a(p^e,m) = (p^(m*e/2)*(p^m+1)-2)/(p^m-1) if e is even else 2*(p^(m*(e+1)/2)-1)/(p^m-1). - Vladeta Jovovic, May 30 2003
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio
R. Sivaramakrishnan, A. Somayajulu, H. Scheid and E. A. Bender, A Number-Theoretic Identity (Advanced Problem 5446 and solutions), American Mathematical Monthly 74 (1967), 1274-1276.
FORMULA
a(n) = Sum_{d divides n} gcd(d, n/d)^2. Multiplicative with a(p^e) = (p^(e+2)+p^e-2)/(p^2-1) if e is even else 2*(p^(e+1)-1)/(p^2-1). - Vladeta Jovovic, May 30 2003
Dirichlet g.f.: zeta^2(s)*zeta(2s-2)/zeta(2s). Dirichlet convolution of A034444 and the sequence n*A010052(n). - R. J. Mathar, Apr 18 2011
Inverse Mobius transform of A008833. - R. J. Mathar, Oct 31 2011
a(n) = Sum_{d divides n} (-1)^A001222(d) * A000010(d) * A000203(n/d) = Sum_{k^2 divides n} k^2 * 2^A001221(n/k^2). - Robert Israel, Oct 18 2015
Sum_{k=1..n} a(k) ~ Zeta(3/2)^2 * n^(3/2) / (3*Zeta(3)) - (3*n*(log(n) - 1 + 2*gamma + 2*log(2*Pi) - 12*Zeta'(2)/Pi^2))/Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 05 2019
a(2^k) = (3/2)*2^k + (1/6)*(-2)^k - 2/3 = A061547(k+2). - Amiram Eldar, Sep 03 2020
MAPLE
R:= proc(n) uses numtheory; local K, k;
K:= select(k -> (n mod k^2 = 0), divisors(n));
add(k^2*2^nops(factorset(n/k^2)), k=K);
end proc:
seq(R(n), n=1..100); # Robert Israel, Oct 18 2015
MATHEMATICA
a[n_]:=Total[GCD[#, n/#]^2 & /@ Divisors[n]]; Table[a[n], {n, 1, 87}] (* Jean-François Alcover, Jul 26 2011 *)
f[p_, e_] := If[OddQ[e], 2*(p^(e+1)-1)/(p^2-1), (p^(e+2)+p^e-2)/(p^2-1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Sep 03 2020 *)
CROSSREFS
KEYWORD
easy,nonn,mult
AUTHOR
Benoit Cloitre, Apr 06 2002
STATUS
approved