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A068454 Factorial expansion of zeta(5) = Sum_{n>=1} a(n)/n!, with a(n) as large as possible. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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

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

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 *)

PROG

(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 log10(n!) exceeds its precision. - M. F. Hasler, Nov 25 2018

(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

(Sage)

b=zeta(5);

def A068454(n, D={}):

    if D.has_key(n):

        return D[n]

    else:

        if (n==1): return floor(b)

        else: return expand(floor(factorial(n)*b) - n*floor(factorial(n-1)*b))

    D[n] = result

    return result

[A068454(n) for n in (1..100)] # G. C. Greubel, Nov 26 2018

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)).

Sequence in context: A117238 A197291 A112983 * A090976 A156199 A135513

Adjacent sequences:  A068451 A068452 A068453 * A068455 A068456 A068457

KEYWORD

nonn

AUTHOR

Benoit Cloitre, Mar 10 2002

EXTENSIONS

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

STATUS

approved

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Last modified August 17 17:25 EDT 2019. Contains 326059 sequences. (Running on oeis4.)