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A256690 From fourth root of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose fourth power is zeta function; sequence gives numerator of b(n).

%I A256690 #30 May 05 2025 09:33:46
%S A256690 1,1,1,5,1,1,1,15,5,1,1,5,1,1,1,195,1,5,1,5,1,1,1,15,5,1,15,5,1,1,1,
%T A256690 663,1,1,1,25,1,1,1,15,1,1,1,5,5,1,1,195,5,5,1,5,1,15,1,15,1,1,1,5,1,
%U A256690 1,5,4641,1,1,1,5,1,1,1,75,1,1,5,5,1,1,1,195,195,1,1,5,1,1,1,15,1,5,1,5,1,1,1,663,1,5,5,25
%N A256690 From fourth root of Riemann zeta function: form Dirichlet series Sum b(n)/n^x whose fourth power is zeta function; sequence gives numerator of b(n).
%C A256690 Dirichlet g.f. of A256690(n)/A256691(n) is (zeta(x))^(1/4).
%C A256690 Formula holds for general Dirichlet g.f. zeta(x)^(1/k) with k = 1, 2, ...
%H A256690 Vaclav Kotesovec, <a href="/A256690/b256690.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..500 from Wolfgang Hintze)
%F A256690 with k = 4;
%F A256690 zeta(x)^(1/k) = Sum_{n>=1} b(n)/n^x;
%F A256690 c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1;
%F A256690 Then solve c(k,n) = 1 for b(m);
%F A256690 a(n) = numerator(b(n)).
%F A256690 Sum_{j=1..n} A256690(j)/A256691(j) ~ n / (Gamma(1/4) * log(n)^(3/4)) * (1 + (3*(1 - gamma/4))/(4*log(n))), where gamma is the Euler-Mascheroni constant A001620 and Gamma() is the gamma function. - _Vaclav Kotesovec_, May 04 2025
%e A256690 b(1), b(2), ... = 1, 1/4, 1/4, 5/32, 1/4, 1/16, 1/4, 15/128, 5/32, 1/16, 1/4, 5/128, 1/4, 1/16, 1/16, 195/2048, ...
%t A256690 k = 4;
%t A256690 c[1, n_] = b[n];
%t A256690 c[k_, n_] := DivisorSum[n, c[1,#1]*c[k - 1, n/#1] & ]
%t A256690 nn = 100; eqs = Table[c[k, n] == 1, {n, 1, nn}];
%t A256690 sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals];
%t A256690 t = Table[b[n], {n, 1, nn}] /. sol[[1]];
%t A256690 num = Numerator[t] (* A256690 *)
%t A256690 den = Denominator[t] (* A256691 *)
%o A256690 (PARI) for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^(1/4))[n]), ", ")) \\ _Vaclav Kotesovec_, May 04 2025
%Y A256690 Cf. A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).
%K A256690 nonn,frac,mult
%O A256690 1,4
%A A256690 _Wolfgang Hintze_, Apr 09 2015