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

1,4

Links

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

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 *)

Crossrefs

Cf. A194762, A194738.

Sequence in context: A279318 A084269 A051427 * A194741 A194753 A359400

Adjacent sequences: A194760 A194761 A194762 * A194764 A194765 A194766

Keyword

nonn,look

Author

Clark Kimberling, Sep 02 2011

Status

approved