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A068976 a(n) = Sum_{d divides n} d/core(d) where core(x) is the smallest number such that x*core(x) is a square. 5

%I

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

%C 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 Robert Israel, <a href="/A068976/b068976.txt">Table of n, a(n) for n = 1..10000</a>

%H Vaclav Kotesovec, <a href="/A068976/a068976.jpg">Graph - the asymptotic ratio</a>

%H 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 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 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 Inverse Mobius transform of A008833. - _R. J. Mathar_, Oct 31 2011

%F 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 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

%p R:= proc(n) uses numtheory; local K,k;

%p K:= select(k -> (n mod k^2 = 0), divisors(n));

%p add(k^2*2^nops(factorset(n/k^2)),k=K);

%p end proc:

%p seq(R(n),n=1..100); # _Robert Israel_, Oct 18 2015

%t a[n_]:=Total[GCD[#, n/#]^2 & /@ Divisors[n]]; Table[a[n], {n, 1, 87}] (* _Jean-Fran├žois Alcover_, Jul 26 2011 *)

%Y Cf. A055155 (m=1).

%Y Cf. A000010, A000203, A001222, A008833, A010052, A034444.

%K easy,nonn,mult

%O 1,2

%A _Benoit Cloitre_, Apr 06 2002

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Last modified April 25 04:14 EDT 2019. Contains 322451 sequences. (Running on oeis4.)