A068454 Factorial expansion of zeta(5) = Sum_{n>=1} a(n)/n!, with a(n) as large as possible.

1, 0, 0, 0, 4, 2, 4, 0, 8, 3, 4, 9, 10, 5, 3, 12, 4, 1, 10, 0, 6, 19, 0, 19, 10, 21, 19, 16, 3, 27, 24, 12, 12, 14, 7, 33, 27, 15, 28, 15, 7, 15, 7, 21, 13, 29, 16, 44, 39, 27, 39, 17, 6, 18, 2, 21, 21, 35, 29, 12, 13, 6, 39, 14, 1, 23, 55, 34, 10, 42, 70, 14, 42, 26, 74, 64, 12, 42, 14

Offset: 1

Author

Benoit Cloitre, Mar 10 2002

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000 (terms 1..300 from Vincenzo Librandi)
• Wikipedia, Factorial number system
• Index entries for factorial base representation of noninteger constants

Crossrefs

Cf. A075874 (same for Pi), A007514 (different variant).

Cf. A067279 (zeta(2)), A067277 (zeta(3)), A068447 (zeta(4)), A068455 (zeta(6)), A068456 (zeta(7)), A068457 (zeta(8)), A068458 (zeta(9)), A068459 (zeta(10)).

Programs

Magma

SetDefaultRealField(RealField(250)); b:=Evaluate(RiemannZeta(),5); [n eq 1 select Floor(b) else Floor(Factorial(n)*b) - n*Floor(Factorial(n)*b/n) : n in [1..100]]; // G. C. Greubel, Nov 26 2018

Mathematica

t = Zeta[5]; s = {}; Do[n = Floor[t*i!]; t -= n/i!; AppendTo[s, n], {i, 1, 30}]; s (* Amiram Eldar, Nov 25 2018 *)
With[{b = Zeta[5]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

PARI

vector(N=100, n, if(n>1, c=c%1*n, c=zeta(precision(5.,N*log(N/2.7)\2.3+3)))\1) \\ Specific a(n) can be computed via the FORMULA. For repeated use the value of c can be stored as a global variable, to be re-computed with higher precision when log_10(n!) exceeds its precision. - M. F. Hasler, Nov 25 2018

Sage

b=zeta(5)
@cached_function
def A068454(n):
    if n == 1: return floor(b)
    else: return expand(floor(factorial(n)*b) - n*floor(factorial(n-1)*b))
[A068454(n) for n in (1..100)] # G. C. Greubel, Nov 26 2018

Formula

a(n) = floor(c*n!) - n*floor(c*(n-1)!) = floor(frac(c*(n-1)!)*n) for n > 1, with c = zeta(5). - M. F. Hasler, Dec 20 2018

Extensions

Name edited and keyword cons removed by M. F. Hasler, Nov 25 2018