A079644 a(n) = (n mod sqrtint(n)).

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

Offset: 1

Author

Benoit Cloitre, Jan 31 2003

Keywords

• nonn
• easy

Comments

Record values: given an m>=0, the first n for which a(n)=m is n = (m+1)^2+m = A028387(m). Also, for n>3, n is a square if and only if a(n)=0 and a(n-1)=0. - Stanislav Sykora, Aug 13 2014

Links

• Stanislav Sykora, Table of n, a(n) for n = 1..1023

Crossrefs

Cf. A000290, A006446, A028387, A033638.

Programs

Maple

a:= proc(n) local r;
r:= isqrt(n);
if r^2 > n then r:= r-1 fi;
n mod r;
end proc:
seq(a(n),n=1..100); # Robert Israel, Aug 13 2014

Mathematica

A079644[n_]:=Mod[n,Floor[n^(1/2)]]; Array[A079644,200] (* Enrique Pérez Herrero, Oct 06 2011 *)
Table[Mod[n,Floor[Sqrt[n]]],{n,110}] (* Harvey P. Dale, Apr 10 2016 *)

PARI

a(n)=n%sqrtint(n)

Formula

a(A006446(n))=0; a(A033638(n))=1.

When n>0, a(A000290(n))=0; when n>1, a(A000290(n)-1)=0. - Stanislav Sykora, Aug 13 2014

a(n) = 0 if n or n+1 or 4*n+1 is a square, otherwise a(n) = a(n-1)+1. - Robert Israel, Aug 13 2014

G.f.: Sum_{r>=2} x^(r^2) * (x^r + 1) * ((r-1)*x^(r+1) - r*x^r + x)/(1 - x)^2. - Robert Israel, Aug 13 2014

Extensions

Definition clarified by N. J. A. Sloane, Jan 11 2025