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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
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

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 *)

CROSSREFS

Cf. A055155 (m=1).

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

Sequence in context: A247765 A129750 A278234 * 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 October 15 20:04 EDT 2019. Contains 328037 sequences. (Running on oeis4.)