|
| |
|
|
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
|
|
|
|
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 log_10(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)
@cached_function
if n == 1: return floor(b)
else: return expand(floor(factorial(n)*b) - n*floor(factorial(n-1)*b))
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
Name edited and keyword cons removed by M. F. Hasler, Nov 25 2018
|
|
|
STATUS
|
approved
|
| |
|
|
