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

%I A068454 #57 Sep 08 2022 08:45:05
%S A068454 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,
%T A068454 27,24,12,12,14,7,33,27,15,28,15,7,15,7,21,13,29,16,44,39,27,39,17,6,
%U A068454 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
%N A068454 Factorial expansion of zeta(5) = Sum_{n>=1} a(n)/n!, with a(n) as large as possible.
%H A068454 G. C. Greubel, <a href="/A068454/b068454.txt">Table of n, a(n) for n = 1..5000</a> (terms 1..300 from Vincenzo Librandi)
%H A068454 Wikipedia, <a href="https://en.wikipedia.org/wiki/Factorial_number_system">Factorial number system</a>
%H A068454 <a href="/index/Fa#facbase">Index entries for factorial base representation of noninteger constants</a>
%F A068454 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
%t A068454 t = Zeta[5]; s = {}; Do[n = Floor[t*i!]; t -= n/i!; AppendTo[s, n], {i, 1, 30}]; s (* _Amiram Eldar_, Nov 25 2018 *)
%t A068454 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 *)
%o A068454 (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
%o A068454 (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
%o A068454 (Sage)
%o A068454 b=zeta(5)
%o A068454 @cached_function
%o A068454 def A068454(n):
%o A068454     if n == 1: return floor(b)
%o A068454     else: return expand(floor(factorial(n)*b) - n*floor(factorial(n-1)*b))
%o A068454 [A068454(n) for n in (1..100)] # _G. C. Greubel_, Nov 26 2018
%Y A068454 Cf. A075874 (same for Pi), A007514 (different variant).
%Y A068454 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)).
%K A068454 nonn
%O A068454 1,5
%A A068454 _Benoit Cloitre_, Mar 10 2002
%E A068454 Name edited and keyword cons removed by _M. F. Hasler_, Nov 25 2018