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A078747
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Expansion of Sum_{k>0} k*phi(k)*x^k/(1+x^k).
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3
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1, 1, 7, 5, 21, 7, 43, 21, 61, 21, 111, 35, 157, 43, 147, 85, 273, 61, 343, 105, 301, 111, 507, 147, 521, 157, 547, 215, 813, 147, 931, 341, 777, 273, 903, 305, 1333, 343, 1099, 441, 1641, 301, 1807, 555, 1281, 507, 2163, 595, 2101, 521, 1911, 785, 2757, 547
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OFFSET
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1,3
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LINKS
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Amiram Eldar, Table of n, a(n) for n = 1..10000
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FORMULA
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Multiplicative with a(2^e) = (4^e-1)/3, a(p^e) = (p^(2*e+1)+1)/(p+1), p>2.
L.g.f.: log(Product_{k>=1} (1 + x^k)^phi(k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 21 2018
Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)/(4*zeta(2)) = 0.182690... (A240976). - Amiram Eldar, Oct 15 2022
Dirichlet g.f.: (zeta(s)*zeta(s-2)/zeta(s-1))*(1-2^(1-s)). - Amiram Eldar, Dec 30 2022
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MATHEMATICA
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f[p_, e_] := If[p == 2, (4^e - 1)/3, (p^(2*e + 1) + 1)/(p + 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Oct 15 2022 *)
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PROG
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(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1] == 2, (4^f[i, 2]-1)/3, (f[i, 1]^(2*f[i, 2]+1)+1)/(f[i, 1]+1))); } \\ Amiram Eldar, Oct 15 2022
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CROSSREFS
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Cf. A000010, A057660, A240976, A299069.
Sequence in context: A204138 A179118 A166639 * A213835 A145396 A263825
Adjacent sequences: A078744 A078745 A078746 * A078748 A078749 A078750
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KEYWORD
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mult,nonn
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AUTHOR
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Vladeta Jovovic, Dec 22 2002
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STATUS
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approved
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