A320471 a(n) = floor(sqrt(n)) mod ceiling(sqrt(n)).

0, 1, 1, 0, 2, 2, 2, 2, 0, 3, 3, 3, 3, 3, 3, 0, 4, 4, 4, 4, 4, 4, 4, 4, 0, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 0, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 0, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 0, 9, 9, 9, 9, 9, 9

Offset: 1

Author

Kritsada Moomuang, Oct 13 2018

Keywords

• nonn

Comments

Sequence consists of zeros interleaved with the positive integers, each positive integer k appearing 2k times.

Crossrefs

Cf. A000196, A003059, A037213, A049240, A173517.

Programs

Magma

[Binomial(Ceiling(Sqrt(n)), Floor(Sqrt(n))) - 1: n in [1..100]]; // Vincenzo Librandi, Dec 02 2018

Maple

a:= proc(n) modp(floor(sqrt(n)),ceil(sqrt(n))) end: seq(a(n),n=1..100); # Muniru A Asiru, Oct 17 2018

Mathematica

Array[Mod[Floor@ #, Ceiling@ #] &@ Sqrt@ # &, 99] (* or *)
Array[IntegerPart@ # - If[IntegerQ@ #, #, 0] &@ Sqrt@ # &, 99] (* or *)
Flatten@ Array[{0}~Join~ConstantArray[#, 2 #] &, 9] (* Michael De Vlieger, Oct 15 2018 *)

PARI

a(n) = sqrtint(n) % (1+sqrtint(n-1)); \\ Michel Marcus, Nov 04 2018

PARI

a(n) = sqrtint(n-1) * !issquare(n) \\ David A. Corneth, Nov 04 2018

Python

from math import isqrt
def A320471(n): return 0 if (m:=isqrt(n))**2==n else m # Chai Wah Wu, Jul 29 2022

Formula

a(n) = A000196(n) - A037213(n).

a(n) = A000196(n)*A049240(n).

a(n) = A000196(n) mod A003059(n).

a(n) = (n - A173517(n)) - A037213(n)^2.

a(n) = binomial(ceiling(sqrt(n)),floor(sqrt(n))) - 1.

From David A. Corneth, Nov 04 2018: (Start)

a(k^2) = 0.

a(m) = floor(sqrt(m)) for nonsquare m. (End)

Extensions

Corrected by Michel Marcus, Jun 14 2022