login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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). 10
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Dirichlet g.f. of A256688(n)/A256689(n) is (zeta (x))^(1/3).

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 = 3;

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, 1/3, 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, ...

MATHEMATICA

k = 3;

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] (* A256688 *)

den = Denominator[t] (* A256689 *)

CROSSREFS

Cf. A046643/A046644 (k=2), A256688/A256689 (k=3), A256690/A256691 (k=4), A256692/A256693 (k=5).

Sequence in context: A236938 A079834 A342458 * A029582 A067095 A070888

Adjacent sequences:  A256685 A256686 A256687 * A256689 A256690 A256691

KEYWORD

nonn,frac,mult

AUTHOR

Wolfgang Hintze, Apr 08 2015

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 17 01:53 EDT 2021. Contains 347478 sequences. (Running on oeis4.)