A013671 - OEIS

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A013671

Decimal expansion of zeta(13).

1, 0, 0, 0, 1, 2, 2, 7, 1, 3, 3, 4, 7, 5, 7, 8, 4, 8, 9, 1, 4, 6, 7, 5, 1, 8, 3, 6, 5, 2, 6, 3, 5, 7, 3, 9, 5, 7, 1, 4, 2, 7, 5, 1, 0, 5, 8, 9, 5, 5, 0, 9, 8, 4, 5, 1, 3, 6, 7, 0, 2, 6, 7, 1, 6, 2, 0, 8, 9, 6, 7, 2, 6, 8, 2, 9, 8, 4, 4, 2, 0, 9, 8, 1, 2, 8, 9, 2, 7, 1, 3, 9, 5, 3, 2, 6, 8, 1, 3

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OFFSET

1,6

REFERENCES

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 811.

LINKS

Table of n, a(n) for n=1..99.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

FORMULA

From Peter Bala, Dec 04 2013: (Start)

Definition: zeta(13) = sum {n >= 1} 1/n^13.

zeta(13) = 2^13/(2^13 - 1)*( sum {n even} n^9*p(n)*p(1/n)/(n^2 - 1)^14 ), where p(n) = n^6 + 21*n^4 + 35*n^2 + 7. (End)

zeta(13) = Sum_{n >= 1} (A010052(n)/n^(13/2)) = Sum_{n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^(13/2) ). - Mikael Aaltonen, Feb 22 2015

zeta(13) = Product_{k>=1} 1/(1 - 1/prime(k)^13). - Vaclav Kotesovec, May 02 2020

EXAMPLE

1.0001227133475784891467518365263573957142751058955098451367026716208967...

MATHEMATICA

RealDigits[Zeta[13], 10, 120][[1]] (* Harvey P. Dale, Dec 24 2016 *)

PROG

(PARI) zeta(13) \\ Charles R Greathouse IV, Apr 25 2016