A065417 - OEIS

%I #29 Jun 13 2020 03:34:59

%S 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,

%T 791,1123,1581,2248,3173,4517,6399,9112,12945,18457,26270,37502,53478,

%U 76416,109146,156135,223301,319764,457884,656288,940795,1349671,1936620

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

%C 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

%H Vaclav Kotesovec, <a href="/A065417/b065417.txt">Table of n, a(n) for n = 1..5000</a>

%H R. J. Mathar, <a href="http://arxiv.org/abs/0903.2514">Hardy-Littlewood constants embedded into infinite products over all positive integers</a>, arXiv:0903.2514 [math.NT], 2009-2011, sequence gamma_{2,j}^(A).

%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a> [Cached copy]

%H <a href="/index/Ar#Artin">Index entries for sequences related to Artin's conjecture</a>

%F a(n) ~ r^n / n, where r = A092526 = 1.465571231876768... - _Vaclav Kotesovec_, Jun 13 2020

%e 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 + ...

%p read("transforms") ;

%p 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:

%p a078012 := [seq(A078012(n),n=1..80)] ; EULERi(%) ;

%p # _R. J. Mathar_, Jul 26 2010

%t A078012[n_] := A078012[n] = If[n<3, {1, 0, 0}[[n+1]], A078012[n-1] + A078012[n-3]]; a078012 = Array[A078012, m = 80];

%t 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_ *)

%Y Cf. A065414.

%K nonn

%O 1,7

%A _N. J. A. Sloane_, Nov 15 2001

%E More terms from _R. J. Mathar_, Jul 26 2010