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A068976
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Sum( d divides n,d/core(d)) where core(x) is the smallest number such that x*core(x) is a square.
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4
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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; internal format)
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OFFSET
| 1,2
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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 (vladeta(AT)eunet.rs), May 30 2003
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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 (vladeta(AT)eunet.rs), 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
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MATHEMATICA
| a[n_]:=Total[GCD[#, n/#]^2 & /@ Divisors[n]]; Table[a[n], {n, 1, 87}] (* From Jean-François Alcover, Jul 26 2011 *)
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CROSSREFS
| Cf. A055155 (m=1)
Sequence in context: A205506 A110141 A129750 * A124859 A021446 A062401
Adjacent sequences: A068973 A068974 A068975 * A068977 A068978 A068979
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KEYWORD
| easy,nonn,mult
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AUTHOR
| Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 06 2002
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