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A007896
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Psi_c(n), where Product_{k>1} 1/(1-1/k^s)^phi(k) = Sum_{k>0} psi_c(k)/k^s.
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10
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1, 1, 2, 3, 4, 4, 6, 7, 9, 8, 10, 12, 12, 12, 16, 18, 16, 19, 18, 24, 24, 20, 22, 32, 30, 24, 34, 36, 28, 40, 30, 42, 40, 32, 48, 60, 36, 36, 48, 64, 40, 60, 42, 60, 76, 44, 46, 86, 63, 66, 64, 72, 52, 82, 80, 96, 72, 56, 58, 128, 60, 60, 114, 104, 96, 100
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OFFSET
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1,3
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COMMENTS
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Phi(k) is the Euler totient function A000010.
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REFERENCES
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Felix Weinstein, The Fibonacci Partitions, preprint, 1995
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LINKS
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EXAMPLE
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The left-hand side (a Dirichlet generating function) is
1/((1-1/2^s)*(1-1/3^s)^2*(1-1/4^s)^2*(1-1/5^s)^4*(1-1/6^s)^2*(1-1/7^s)^6* ...)
= 1 + 1/2^s + 2/3^s + 3/4^s + 4/5^s + 4/6^s + 6/7^s + 7/8^s + 9/9^s + ...,
whose coefficients are 1, 1, 2, 3, 4, 4, 6, 7, 9, ... . - N. J. A. Sloane, May 26 2014
G.f. = x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 4*x^6 + 6*x^7 + 7*x^8 + 9*x^9 + ...
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MATHEMATICA
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dircon[v_, w_] := Module[{lv = Length[v], lw = Length[w], fv, fw}, fv[n_] := If[n <= lv, v[[n]], 0]; fw[n_] := If[n <= lw, w[[n]], 0]; Table[ DirichletConvolve[fv[n], fw[n], n, m], {m, Min[lv, lw]}]];
a[n_] := Module[{A, v, w, m}, If[n<1, 0, v = Table[Boole[k == 1], {k, n}]; For[k = 2, k <= n, k++, m = Length[IntegerDigits[n, k]] - 1; A = (1 - x)^-EulerPhi[k] + x*O[x]^m // Normal; w = Table[0, {n}]; For[i = 0, i <= m, i++, w[[k^i]] = Coefficient[A, x, i]]; v = dircon[v, w]]; v[[n]]]];
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PROG
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(PARI) {a(n) = my(A, v, w, m); if( n<1, 0, v = vector(n, k, k==1); for(k=2, n, m = #digits(n, k) - 1; A = (1 - x)^ -eulerphi(k) + x * O(x^m); w = vector(n); for(i=0, m, w[k^i] = polcoeff(A, i)); v = dirmul(v, w)); v[n])}; /* Michael Somos, May 26 2014 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Felix Weinstein (wain(AT)ana.unibe.ch)
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EXTENSIONS
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Definition corrected by Felix Weinstein (wain(AT)ana.unibe.ch), May 14 2014
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STATUS
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approved
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