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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)).

%I A065417 #29 Jun 13 2020 03:34:59
%S A065417 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 A065417 791,1123,1581,2248,3173,4517,6399,9112,12945,18457,26270,37502,53478,
%U A065417 76416,109146,156135,223301,319764,457884,656288,940795,1349671,1936620
%N 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)).
%C A065417 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 A065417 Vaclav Kotesovec, <a href="/A065417/b065417.txt">Table of n, a(n) for n = 1..5000</a>
%H A065417 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 A065417 G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a> [Cached copy]
%H A065417 <a href="/index/Ar#Artin">Index entries for sequences related to Artin's conjecture</a>
%F A065417 a(n) ~ r^n / n, where r = A092526 = 1.465571231876768... - _Vaclav Kotesovec_, Jun 13 2020
%e A065417 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 A065417 read("transforms") ;
%p A065417 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 A065417 a078012 := [seq(A078012(n),n=1..80)] ; EULERi(%) ;
%p A065417 # _R. J. Mathar_, Jul 26 2010
%t A065417 A078012[n_] := A078012[n] = If[n<3, {1, 0, 0}[[n+1]], A078012[n-1] + A078012[n-3]]; a078012 = Array[A078012, m = 80];
%t A065417 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 A065417 Cf. A065414.
%K A065417 nonn
%O A065417 1,7
%A A065417 _N. J. A. Sloane_, Nov 15 2001
%E A065417 More terms from _R. J. Mathar_, Jul 26 2010