A166248 a(n) is the absolute value of n minus sum of all the remainders modulo the numbers below n.

1, 2, 2, 3, 1, 3, 1, 0, 3, 3, 11, 5, 15, 17, 21, 20, 34, 29, 45, 41, 49, 55, 75, 61, 78, 86, 98, 96, 122, 108, 136, 135, 151, 163, 183, 162, 196, 210, 230, 218, 256, 242, 282, 284, 294, 312, 356, 326, 365, 370, 398, 402, 452, 438, 474, 464, 496, 520, 576, 526, 584, 610

Offset: 1

Author

Juri-Stepan Gerasimov, Oct 10 2009

Keywords

• nonn

Examples

			a(1) = abs(1-0) = 1;
a(2) = abs(2-0) = 2;
a(3) = abs(3-1) = 2;
a(4) = abs(4-1) = 3;
a(5) = abs(5-4) = 1;
a(6) = abs(6-3) = 3;
a(7) = abs(7-8) = 1.
		

Crossrefs

Cf. A000027, A004125, A163180.

Programs

Maple

A004125 := proc(n) add( modp(n,k),k=1..n) ; end proc: A166248 := proc(n) abs(n-A004125(n)) ; end: seq(A166248(n),n=1..100) ; # R. J. Mathar, Oct 24 2009

Python

from math import isqrt
def A166248(n): return abs(n*(n-1)+((s:=isqrt(n))**2*(s+1)-sum((q:=n//k)*((k<<1)+q+1) for k in range(1,s+1))>>1)) # Chai Wah Wu, Nov 01 2023

Formula

a(n) = abs(n - Sum_{k=1..n} (n mod k)).

a(n) = abs(n - A004125(n)). - Michel Marcus, May 08 2019

Extensions

a(19), a(20), a(37) etc. corrected by R. J. Mathar, Oct 24 2009