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A256689 From third root of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose cube is zeta function; sequence gives denominator of b(n).

%I A256689 #42 May 05 2025 09:33:19
%S A256689 1,3,3,9,3,9,3,81,9,9,3,27,3,9,9,243,3,27,3,27,9,9,3,243,9,9,81,27,3,
%T A256689 27,3,729,9,9,9,81,3,9,9,243,3,27,3,27,27,9,3,729,9,27,9,27,3,243,9,
%U A256689 243,9,9,3,81,3,9,27,6561,9,27,3,27,9,27,3,729,3,9,27,27,9,27,3,729,243,9,3,81,9,9,9,243,3,81,9,27,9,9,9,2187,3,27,27,81
%N A256689 From third root of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose cube is zeta function; sequence gives denominator of b(n).
%C A256689 Dirichlet g.f. of A256688(n)/A256689(n) is (zeta(x))^(1/3).
%C A256689 Formula holds for general Dirichlet g.f. zeta(x)^(1/k) with k = 1, 2, ...
%H A256689 Vaclav Kotesovec, <a href="/A256689/b256689.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..500 from Wolfgang Hintze)
%F A256689 with k = 3;
%F A256689 zeta(x)^(1/k) = Sum_{n>=1} b(n)/n^x;
%F A256689 c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
%F A256689 Then solve c(k,n) = 1 for b(m);
%F A256689 a(n) = denominator(b(n)).
%F A256689 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 A256689 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 A256689 k = 3;
%t A256689 c[1, n_] = b[n];
%t A256689 c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ]
%t A256689 nn = 100; eqs = Table[c[k, n] == 1, {n, 1, nn}];
%t A256689 sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals];
%t A256689 t = Table[b[n], {n, 1, nn}] /. sol[[1]];
%t A256689 num = Numerator[t] (* A256688 *)
%t A256689 den = Denominator[t] (* A256689 *)
%o A256689 (PARI) for(n=1, 100, print1(denominator(direuler(p=2, n, 1/(1-X)^(1/3))[n]), ", ")) \\ _Vaclav Kotesovec_, May 04 2025
%Y A256689 Cf. A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).
%K A256689 nonn,frac,mult
%O A256689 1,2
%A A256689 _Wolfgang Hintze_, Apr 08 2015