A007896 Psi_c(n), where Product_{k>1} 1/(1-1/k^s)^phi(k) = Sum_{k>0} psi_c(k)/k^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, 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

Offset: 1

Author

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

Keywords

• nonn

Comments

Phi(k) is the Euler totient function A000010.

Examples

			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 + ...
		

References

• Felix Weinstein, The Fibonacci Partitions, preprint, 1995

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..1000
• F. V. Weinstein, Notes on Fibonacci partitions, arXiv:math/0307150 [math.NT], 2003-2015.

Crossrefs

Cf. A000010, A007897, A007898.

Programs

Mathematica

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]]]];
Array[a, 66] (* Jean-François Alcover, Nov 12 2018, from PARI *)

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 */

Extensions

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