A013672 - OEIS

A013672

Decimal expansion of zeta(14).

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

(list; constant; graph; refs; listen; history; text; internal format)

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.

Index entries for transcendental numbers.

Index entries for constants related to zeta.

FORMULA

zeta(14) = Sum_{n >= 1} (A010052(n)/n^7) = Sum {n >= 1} ( (floor(sqrt(n))-floor(sqrt(n-1)))/n^7 ). - Mikael Aaltonen, Feb 20 2015

zeta(14) = (2/18243225)*Pi^14 (see A046988/A002432). - Rick L. Shepherd, May 30 2016

zeta(14) = Product_{k>=1} 1/(1 - 1/prime(k)^14). - Vaclav Kotesovec, May 02 2020

From Andrea Pinos, May 06 2026: (Start)

zeta(14) = Sum_{k=1..(n-1)} ((Gamma(k/n)*Gamma(1 - k/n))^14) / (2*(n^14 + (691/30)*n^12 + 273*n^10 + (44473/20)*n^8 + (42328/3)*n^6 + (386568/5)*n^4 + 518400*n^2 - 36740617/60)).

More generally: Sum_{k=1..(n-1)} ((Gamma(k/n)*Gamma(1 - k/n))^p) = 2*(zeta(p)*n^p + p*zeta(2)*zeta(p-2)*n^(p-2) + (5*p^2/4 + p/2)*zeta(4)*zeta(p-4)*n^(p-4) + (35*p^3/48 + 7*p^2/8 + p/3)*zeta(6)*zeta(p-6)*n^(p-6) + (175*p^4/576 + 35*p^3/48 + 101*p^2/144 + p/4)*zeta(8)*zeta(p-8)*n^(p-8) + ... ). (End)

EXAMPLE

1.0000612481350587048292585451051353337474816961691545494827552022528629...

MATHEMATICA

RealDigits[Zeta[14], 10, 120][[1]] (* Harvey P. Dale, Dec 19 2014 *)

PROG

(PARI) zeta(14) \\ Michel Marcus, Feb 20 2015

CROSSREFS

Cf. A010052, A002432, A046988.

Sequence in context: A083463 A187110 A378994 * A019946 A090551 A220782

Adjacent sequences: A013669 A013670 A013671 * A013673 A013674 A013675

KEYWORD

nonn,cons

AUTHOR

N. J. A. Sloane

STATUS

approved