cp's OEIS Frontend

A248227 Least k such that zeta(4) - sum{1/h^4, h = 1..k} < 1/n^3.

%I A248227 #9 Oct 09 2014 07:13:15
%S A248227 1,1,2,3,3,4,5,6,6,7,8,8,9,10,10,11,12,12,13,14,15,15,16,17,17,18,19,
%T A248227 19,20,21,21,22,23,24,24,25,26,26,27,28,28,29,30,31,31,32,33,33,34,35,
%U A248227 35,36,37,37,38,39,40,40,41,42,42,43,44,44,45,46,46,47
%N A248227 Least k such that zeta(4) - sum{1/h^4, h = 1..k} < 1/n^3.
%C A248227 This sequence and A248230 provide insight into the manner of convergence of sum{1/h^4, h = 0..k}.  Since a(n+1) - a(n) is in {0,1} for n >= 1, A248228 and A248229 are complementary.
%H A248227 Clark Kimberling, <a href="/A248227/b248227.txt">Table of n, a(n) for n = 1..1000</a>
%F A248227 a(n) ~ 3^(-1/3) * n. - _Vaclav Kotesovec_, Oct 09 2014
%e A248227 Let s(n) = sum{1/h^4, h = 1..n}.  Approximations are shown here:
%e A248227 n ... zeta(4) - s(n) ... 1/n^3
%e A248227 1 ... 0.0823232 .... 1
%e A248227 2 ... 0.0198232 .... 0.125
%e A248227 3 ... 0.0074775 .... 0.037
%e A248227 4 ... 0.0035713 .... 0.015
%e A248227 5 ... 0.0019713 .... 0.008
%e A248227 6 ... 0.0011997 .... 0.005
%e A248227 a(6) = 4 because zeta(4) - s(4) < 1/216 < zeta(4) - s(3).
%t A248227 $MaxExtraPrecision = Infinity; z = 400; p[k_] := p[k] = Sum[1/h^4, {h, 1, k}];
%t A248227 N[Table[Zeta[4] - p[n], {n, 1, z/10}]]
%t A248227 f[n_] := f[n] = Select[Range[z], Zeta[4] - p[#] < 1/n^3 &, 1];
%t A248227 u = Flatten[Table[f[n], {n, 1, z}]]   (* A248227 *)
%t A248227 Flatten[Position[Differences[u], 0]]  (* A248228 *)
%t A248227 Flatten[Position[Differences[u], 1]]  (* A248229 *)
%t A248227 f = Table[Floor[1/(Zeta[4] - p[n])], {n, 1, z}]  (* A248230 *)
%Y A248227 Cf. A248228, A248229, A248230, A013662.
%K A248227 nonn,easy
%O A248227 1,3
%A A248227 _Clark Kimberling_, Oct 05 2014