A067279 -id:A067279 - cp's OEIS frontend

A067277 Factorial expansion of zeta(3): zeta(3) = Sum_{n>=1} a(n)/n!.

1, 0, 1, 0, 4, 1, 3, 2, 8, 4, 0, 11, 11, 10, 9, 4, 2, 11, 5, 12, 16, 12, 6, 3, 22, 22, 12, 14, 23, 1, 24, 24, 12, 14, 1, 27, 14, 26, 21, 16, 22, 14, 6, 19, 12, 12, 36, 22, 32, 38, 10, 1, 14, 51, 9, 6, 51, 26, 50, 25, 30, 44, 19, 49, 12, 17, 24, 55, 17, 47, 11, 8, 43, 71, 43, 16, 76

Offset: 1

Benoit Cloitre, Mar 10 2002

			zeta(3) = 1 + 1/3! + 4/5! + 1/6! + 3/7! + 2/8! + 8/9! + 4/10! + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A067279 (zeta(2)), A068447 (zeta(4)), A068454 (zeta(5)), A068455 (zeta(6)), A068456 (zeta(7)), A068457 (zeta(8)), A068458 (zeta(9)), A068459 (zeta(10)).

SetDefaultRealField(RealField(250)); L:=RiemannZeta(); [Floor(Evaluate(L,3))] cat [Floor(Factorial(n)*Evaluate(L,3)) - n*Floor(Factorial((n-1))*Evaluate(L,3)) : n in [2..80]]; // G. C. Greubel, Nov 26 2018

With[{b = Zeta[3]}, Table[If[n == 1, Floor[b], Floor[n!*b] - n*Floor[(n - 1)!*b]], {n, 1, 100}]] (* G. C. Greubel, Nov 26 2018 *)

default(realprecision, 250); b = zeta(3); for(n=1, 80, print1(if(n==1, floor(b), floor(n!*b) - n*floor((n-1)!*b)), ", ")) \\ G. C. Greubel, Nov 26 2018

def A067279(n):
    if (n==1): return floor(zeta(3))
    else: return expand(floor(factorial(n)*zeta(3)) - n*floor(factorial(n-1)*zeta(3)))
[A067279(n) for n in (1..80)] # G. C. Greubel, Nov 26 2018

a(n) = floor(n!*zeta(3)) - n*floor((n-1)!*zeta(3)), with a(1)=1, for n > 1.

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

Original entry on oeis.org

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

Benoit Cloitre, Mar 10 2002

nonn

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

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)).

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

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 *)

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

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

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

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

cp's OEIS Frontend

A067277 Factorial expansion of zeta(3): zeta(3) = Sum_{n>=1} a(n)/n!.

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