A065417 Exponents in expansion of rank-2 Artin constant product(1-1/(p^3-p^2), p=prime) as a product zeta(n)^(-a(n)).

0, 0, 1, 1, 1, 1, 2, 2, 3, 4, 6, 7, 11, 14, 20, 27, 39, 52, 75, 102, 145, 201, 286, 397, 565, 791, 1123, 1581, 2248, 3173, 4517, 6399, 9112, 12945, 18457, 26270, 37502, 53478, 76416, 109146, 156135, 223301, 319764, 457884, 656288, 940795, 1349671, 1936620

Offset

1,7

Comments

Inverse Euler transform of A078012. (The inverse of 1-1/(p^3-p^2) is p^2(p-1)/(p^3-p^2-1) = 1-1/(1+p^2-p^3). Setting 1/p=x gives (1-x)/(1-x-x^3), the g.f. of A078012.) - R. J. Mathar, Jul 26 2010

Links

Vaclav Kotesovec, Table of n, a(n) for n = 1..5000

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, sequence gamma_{2,j}^(A).

G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]

Index entries for sequences related to Artin's conjecture

Formula

a(n) ~ r^n / n, where r = A092526 = 1.465571231876768... - Vaclav Kotesovec, Jun 13 2020

Example

x^3 + x^4 + x^5 + x^6 + 2*x^7 + 2*x^8 + 3*x^9 + 4*x^10 + 6*x^11 + 7*x^12 + ...

Maple

read("transforms") ;
A078012 := proc(n) option remember; if n <3 then op(n+1, [1, 0, 0]) ; else procname(n-1)+procname(n-3) ; end if; end proc:
a078012 := [seq(A078012(n), n=1..80)] ; EULERi(%) ;
# R. J. Mathar, Jul 26 2010

Mathematica

A078012[n_] := A078012[n] = If[n<3, {1, 0, 0}[[n+1]], A078012[n-1] + A078012[n-3]]; a078012 = Array[A078012, m = 80];
s = {}; For[i = 1, i <= m, i++, AppendTo[s, i*a078012[[i]] - Sum[s[[d]] * a078012[[i-d]], {d, i-1}]]]; Table[Sum[If[Divisible[i, d], MoebiusMu[i/d ], 0]*s[[d]], {d, 1, i}]/i, {i, m}] (* Jean-François Alcover, Apr 15 2016, after R. J. Mathar *)

Crossrefs

Cf. A065414.

Sequence in context: A274156 A094860 A213816 * A005860 A266900 A114541

Adjacent sequences: A065414 A065415 A065416 * A065418 A065419 A065420

Keyword

nonn

Author

N. J. A. Sloane, Nov 15 2001

Extensions

More terms from R. J. Mathar, Jul 26 2010

Status

approved