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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.
10

%I #54 Nov 12 2018 11:13:40

%S 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,

%T 34,36,28,40,30,42,40,32,48,60,36,36,48,64,40,60,42,60,76,44,46,86,63,

%U 66,64,72,52,82,80,96,72,56,58,128,60,60,114,104,96,100

%N Psi_c(n), where Product_{k>1} 1/(1-1/k^s)^phi(k) = Sum_{k>0} psi_c(k)/k^s.

%C Phi(k) is the Euler totient function A000010.

%D Felix Weinstein, The Fibonacci Partitions, preprint, 1995

%H N. J. A. Sloane, <a href="/A007896/b007896.txt">Table of n, a(n) for n = 1..1000</a>

%H F. V. Weinstein, <a href="http://arXiv.org/abs/math.NT/0307150">Notes on Fibonacci partitions</a>, arXiv:math/0307150 [math.NT], 2003-2015.

%e The left-hand side (a Dirichlet generating function) is

%e 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* ...)

%e = 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 + ...,

%e whose coefficients are 1, 1, 2, 3, 4, 4, 6, 7, 9, ... . - _N. J. A. Sloane_, May 26 2014

%e 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 + ...

%t 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]}]];

%t 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]]]];

%t Array[a, 66] (* _Jean-François Alcover_, Nov 12 2018, from PARI *)

%o (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 */

%Y Cf. A000010, A007897, A007898.

%K nonn

%O 1,3

%A Felix Weinstein (wain(AT)ana.unibe.ch)

%E Definition corrected by Felix Weinstein (wain(AT)ana.unibe.ch), May 14 2014