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

1,5

Comments

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

Links

Table of n, a(n) for n=1..87.

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)

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 *)

Prog

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

Crossrefs

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

Sequence in context: A305629 A214664 A214666 * A333180 A127444 A241477

Adjacent sequences: A320468 A320469 A320470 * A320472 A320473 A320474

Keyword

nonn

Author

Kritsada Moomuang, Oct 13 2018

Extensions

Corrected by Michel Marcus, Jun 14 2022

Status

approved