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A068976 a(n) = Sum_{d | n} d/core(d) where core(x) is the smallest number such that x*core(x) is a square.

%I A068976 #47 Sep 03 2020 09:50:42
%S A068976 1,2,2,6,2,4,2,10,11,4,2,12,2,4,4,26,2,22,2,12,4,4,2,20,27,4,20,12,2,
%T A068976 8,2,42,4,4,4,66,2,4,4,20,2,8,2,12,22,4,2,52,51,54,4,12,2,40,4,20,4,4,
%U A068976 2,24,2,4,22,106,4,8,2,12,4,8,2,110,2,4,54,12,4,8,2,52,101,4,2,24,4,4,4
%N A068976 a(n) = Sum_{d | n} d/core(d) where core(x) is the smallest number such that x*core(x) is a square.
%C A068976 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
%H A068976 Robert Israel, <a href="/A068976/b068976.txt">Table of n, a(n) for n = 1..10000</a>
%H A068976 Vaclav Kotesovec, <a href="/A068976/a068976.jpg">Graph - the asymptotic ratio</a>
%H A068976 R. Sivaramakrishnan, A. Somayajulu, H. Scheid and E. A. Bender, <a href="http://dx.doi.org/10.2307/2315711">A Number-Theoretic Identity (Advanced Problem 5446 and solutions)</a>, American Mathematical Monthly 74 (1967), 1274-1276.
%F A068976 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
%F A068976 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
%F A068976 Inverse Mobius transform of A008833. - _R. J. Mathar_, Oct 31 2011
%F A068976 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
%F A068976 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
%F A068976 a(2^k) = (3/2)*2^k + (1/6)*(-2)^k - 2/3 = A061547(k+2). - _Amiram Eldar_, Sep 03 2020
%p A068976 R:= proc(n) uses numtheory; local K,k;
%p A068976   K:= select(k -> (n mod k^2 = 0), divisors(n));
%p A068976   add(k^2*2^nops(factorset(n/k^2)),k=K);
%p A068976 end proc:
%p A068976 seq(R(n),n=1..100); # _Robert Israel_, Oct 18 2015
%t A068976 a[n_]:=Total[GCD[#, n/#]^2 & /@ Divisors[n]]; Table[a[n], {n, 1, 87}] (* _Jean-François Alcover_, Jul 26 2011 *)
%t A068976 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 *)
%Y A068976 Cf. A055155 (m=1).
%Y A068976 Cf. A000010, A000203, A001222, A008833, A010052, A034444, A061547.
%K A068976 easy,nonn,mult
%O A068976 1,2
%A A068976 _Benoit Cloitre_, Apr 06 2002