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A194763 Number of k < n such that {k*2^(1/3)} > {n*2^(1/3)}, where { } = fractional part.

%I A194763 #11 Jan 31 2025 11:57:17
%S A194763 0,0,0,3,2,1,0,6,4,2,0,9,6,3,0,12,8,4,0,15,10,5,0,18,12,6,26,19,12,5,
%T A194763 28,20,12,4,30,21,12,3,32,22,12,2,34,23,12,1,36,24,12,0,38,25,12,52,
%U A194763 38,24,10,53,38,23,8,54,38,22,6,55,38,21,4,56,38,20,2,57,38,19,76
%N A194763 Number of k < n such that {k*2^(1/3)} > {n*2^(1/3)}, where { } = fractional part.
%H A194763 Robert Israel, <a href="/A194763/b194763.txt">Table of n, a(n) for n = 1..10000</a>
%p A194763 N:= 100: # for a(1) .. a(N)
%p A194763 S:= [seq(frac(k*2^(1/3)),k=1..N)]:
%p A194763 compare:= proc(x,y) local z,a,b;
%p A194763   z:= y - x;
%p A194763   a:= coeff(z,2^(1/3));
%p A194763   b:= z - a*2^(1/3);
%p A194763   2*a^3 + b^3 > 0
%p A194763 end proc:
%p A194763 seq(nops(select(t -> compare(S[n],t),S[1..n-1])), n=1..N); # _Robert Israel_, Jan 31 2025
%t A194763 r = 2^(1/3); p[x_] := FractionalPart[x];
%t A194763 u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
%t A194763 v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
%t A194763 s[n_] := Sum[u[n, k], {k, 1, n}]
%t A194763 t[n_] := Sum[v[n, k], {k, 1, n}]
%t A194763 Table[s[n], {n, 1, 100}]  (* A194762 *)
%t A194763 Table[t[n], {n, 1, 100}]  (* A194763 *)
%Y A194763 Cf. A194762, A194738.
%K A194763 nonn,look
%O A194763 1,4
%A A194763 _Clark Kimberling_, Sep 02 2011