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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). 10
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Dirichlet g.f. of A256690(n)/A256691(n) is (zeta (x))^(1/4). Formula holds for general Dirichlet g.f. zeta(x)^(1/k) with k = 1, 2, ... LINKS Wolfgang Hintze, Table of n, a(n) for n = 1..500 FORMULA with k = 4; zeta(x)^(1/k) = Sum_{n>=1} b(n)/n^x; c(1,n)=b(n); c(k,n) = Sum_{d|n} c(1,d)*c(k-1,n/d), k>1; Then solve c(k,n) = 1 for b(m); a(n) = numerator(b(n)). EXAMPLE 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, ... MATHEMATICA k = 4; c[1, n_] = b[n]; c[k_, n_] := DivisorSum[n, c[1, #1]*c[k - 1, n/#1] & ] nn = 100; eqs = Table[c[k, n] == 1, {n, 1, nn}]; sol = Solve[Join[{b[1] == 1}, eqs], Table[b[i], {i, 1, nn}], Reals]; t = Table[b[n], {n, 1, nn}] /. sol[[1]]; num = Numerator[t] (* A256690 *) den = Denominator[t] (* A256691 *) CROSSREFS Cf. A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5). Sequence in context: A353958 A285486 A230368 * A181985 A304320 A130511 Adjacent sequences:  A256687 A256688 A256689 * A256691 A256692 A256693 KEYWORD nonn,frac,mult AUTHOR Wolfgang Hintze, Apr 09 2015 STATUS approved

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Last modified August 18 10:08 EDT 2022. Contains 356204 sequences. (Running on oeis4.)