A347428 Expansion of g.f. Product_{k>=2} 1/(1-x^phi(k)).

1, 1, 4, 4, 14, 14, 40, 40, 106, 106, 254, 254, 582, 582, 1256, 1256, 2620, 2620, 5256, 5256, 10266, 10266, 19482, 19482, 36204, 36204, 65792, 65792, 117496, 117496, 206120, 206120, 356320, 356320, 606912, 606912, 1020848, 1020848, 1695676, 1695676, 2786010

Offset: 0

Author

Michel Marcus, Sep 02 2021

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• David P. Roberts and Fernando Rodriguez Villegas, Hypergeometric Motives, arXiv:2109.00027 [math.AG], 2021. See (5.2) p. 6.

Crossrefs

Cf. A000010 (phi), A014197, A051894, A120963 (similar g.f.).

Programs

Maple

with(numtheory):
b:= proc(n) option remember; nops(invphi(n)) end:
g:= proc(n) option remember; `if`(n=0, 1, add(
      g(n-j)*add(d*b(d), d=divisors(j)), j=1..n)/n)
    end:
a:= n-> g(n)-g(n-1):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 23 2023

Mathematica

nt = 100; (* number of terms *)
f[kmax_] := f[kmax] = CoefficientList[Product[1/(1 - x^EulerPhi[k]), {k, 2, kmax}] + O[x]^nt, x]; f[kmax = nt]; f[kmax += nt];
While[f[kmax] != f[kmax - nt], kmax += nt];
f[kmax] (* Jean-François Alcover, Nov 29 2023 *)

Formula

From Vaclav Kotesovec, Sep 02 2021: (Start)

For n>0, a(n) = A120963(n) - A120963(n-1).

log(a(n)) ~ sqrt(105*zeta(3)*n)/Pi. (End)

Extensions

Terms a(16) and beyond corrected by Vaclav Kotesovec, Jun 23 2023, following a suggestion from Georg Fischer