cp's OEIS Frontend

A256688 From third root of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose cube is zeta function; sequence gives numerator of b(n).

%I A256688 #47 May 05 2025 16:38:34
%S A256688 1,1,1,2,1,1,1,14,2,1,1,2,1,1,1,35,1,2,1,2,1,1,1,14,2,1,14,2,1,1,1,91,
%T A256688 1,1,1,4,1,1,1,14,1,1,1,2,2,1,1,35,2,2,1,2,1,14,1,14,1,1,1,2,1,1,2,
%U A256688 728,1,1,1,2,1,1,1,28,1,1,2,2,1,1,1,35,35,1,1,2,1,1,1,14,1,2,1,2,1,1,1,91,1,2,2,4
%N A256688 From third root of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose cube is zeta function; sequence gives numerator of b(n).
%C A256688 Dirichlet g.f. of A256688(n)/A256689(n) is (zeta(x))^(1/3).
%C A256688 Formula holds for general Dirichlet g.f. zeta(x)^(1/k) with k = 1, 2, ...
%H A256688 Vaclav Kotesovec, <a href="/A256688/b256688.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..500 from Wolfgang Hintze)
%F A256688 with k = 3;
%F A256688 zeta(x)^(1/k) = Sum_{n>=1} b(n)/n^x;
%F A256688 c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
%F A256688 Then solve c(k,n) = 1 for b(m);
%F A256688 a(n) = numerator(b(n)).
%F A256688 Sum_{j=1..n} A256688(j)/A256689(j) ~ n / (Gamma(1/3) * log(n)^(2/3)) * (1 + (2*(1 - gamma/3))/(3*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function. - _Vaclav Kotesovec_, May 04 2025
%e A256688 b(1), b(2), ... = 1, 1/3, 1/3, 2/9, 1/3, 1/9, 1/3, 14/81, 2/9, 1/9, 1/3, 2/27, 1/3, 1/9, 1/9, 35/243, ...
%t A256688 k = 3;
%t A256688 c[1, n_] = b[n];
%t A256688 c[k_, n_] := DivisorSum[n, c[1,#1]*c[k - 1, n/#1] & ]
%t A256688 nn = 100; eqs = Table[c[k, n] == 1, {n, 1, nn}];
%t A256688 sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals];
%t A256688 t = Table[b[n], {n, 1, nn}] /. sol[[1]];
%t A256688 num = Numerator[t] (* A256688 *)
%t A256688 den = Denominator[t] (* A256689 *)
%o A256688 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(1/3))[n]), ", ")) \\ _Vaclav Kotesovec_, May 04 2025
%Y A256688 Cf. A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).
%K A256688 nonn,frac,mult
%O A256688 1,4
%A A256688 _Wolfgang Hintze_, Apr 08 2015