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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. 5
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 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 Cf. A055155 (m=1). Cf. A000010, A000203, A001222, A008833, A010052, A034444, A061547. Sequence in context: A129750 A278234 A349330 * A265392 A253139 A318519 Adjacent sequences:  A068973 A068974 A068975 * A068977 A068978 A068979 KEYWORD easy,nonn,mult AUTHOR Benoit Cloitre, Apr 06 2002 STATUS approved

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Last modified January 27 22:53 EST 2022. Contains 350654 sequences. (Running on oeis4.)