A056847 Nearest integer to n - sqrt(n).

0, 0, 1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59

Offset: 0

Author

N. J. A. Sloane, Dec 07 2000

Keywords

• nonn

References

• B. Alspach, K. Heinrich and G. Liu, Orthogonal factorizations of graphs, pp. 13-40 of Contemporary Design Theory, ed. J. H. Dinizt and D. R. Stinson, Wiley, 1992 (see Theorem 2.7).

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000194, A028391.

Programs

Magma

[n-Floor(Sqrt(n)+1/2):n in [0..80]]; // Marius A. Burtea, May 13 2019

Maple

0,seq(seq(n-k, n=k^2-k+1..k^2+k),k=1..10); # Robert Israel, Jun 13 2018

Mathematica

Table[Round[n-Sqrt[n]],{n,0,70}] (* Harvey P. Dale, Jun 15 2014 *)

PARI

a(n) = round(n - sqrt(n)); \\ Michel Marcus, May 13 2019

Python

from math import isqrt
def A056847(n): return n-(m:=isqrt(n))-int(n>m*(m+1)) # Chai Wah Wu, Jun 05 2025

Formula

From Robert Israel, Jun 13 2018: (Start)

a(n) = n-k for k^2-k+1 <= n <= k^2+k, k >= 1.

G.f.: x/(1-x)^2 - Theta_2(0,x)*x^(3/4)/(2*(1-x)) where Theta_2 is a Jacobi theta function. (End)

a(n) = n - floor(sqrt(n) + 1/2) = n - A000194(n). - Ridouane Oudra, May 13 2019