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

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, 28, 20, 12, 4, 30, 21, 12, 3, 32, 22, 12, 2, 34, 23, 12, 1, 36, 24, 12, 0, 38, 25, 12, 52, 38, 24, 10, 53, 38, 23, 8, 54, 38, 22, 6, 55, 38, 21, 4, 56, 38, 20, 2, 57, 38, 19, 76

Offset: 1

Author

Clark Kimberling, Sep 02 2011

Keywords

• nonn
• look

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A194762, A194738.

Programs

Maple

N:= 100: # for a(1) .. a(N)
S:= [seq(frac(k*2^(1/3)),k=1..N)]:
compare:= proc(x,y) local z,a,b;
  z:= y - x;
  a:= coeff(z,2^(1/3));
  b:= z - a*2^(1/3);
  2*a^3 + b^3 > 0
end proc:
seq(nops(select(t -> compare(S[n],t),S[1..n-1])), n=1..N); # Robert Israel, Jan 31 2025

Mathematica

r = 2^(1/3); p[x_] := FractionalPart[x];
u[n_, k_] := If[p[k*r] <= p[n*r], 1, 0]
v[n_, k_] := If[p[k*r] > p[n*r], 1, 0]
s[n_] := Sum[u[n, k], {k, 1, n}]
t[n_] := Sum[v[n, k], {k, 1, n}]
Table[s[n], {n, 1, 100}]  (* A194762 *)
Table[t[n], {n, 1, 100}]  (* A194763 *)