A340740 a(n) is the sum of all the remainders when n is divided by positive integers less than n/2 and coprime to n.

0, 0, 0, 0, 1, 0, 2, 2, 2, 1, 7, 2, 7, 6, 5, 4, 15, 7, 19, 10, 9, 8, 32, 9, 20, 20, 28, 13, 46, 14, 47, 31, 27, 31, 48, 17, 62, 39, 58, 26, 87, 26, 94, 53, 52, 41, 127, 48, 100, 65, 79, 61, 154, 52, 105, 62, 90, 80, 200, 45, 180, 113, 138, 103, 162, 77, 229, 116, 149, 73, 274, 87, 257, 166, 178

Offset: 1

Author

J. M. Bergot and Robert Israel, Jan 18 2021

Keywords

• nonn
• look

Examples

			For n = 11, a(11) = (11 mod 1)+(11 mod 2)+(11 mod 3)+(11 mod 4)+(11 mod 5) = 7.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A067439.

Programs

Maple

f:= proc(n) local k;
  add(`if`(igcd(k,n)=1, n mod k, 0),k=1..floor(n/2))
end proc:
map(f, [$1..100]);

Mathematica

Table[Sum[Mod[n, i]*Floor[1/GCD[i, n]], {i, Floor[(n - 1)/2]}], {n,
100}] (* Wesley Ivan Hurt, Jan 18 2021 *)

PARI

a(n) = sum(k=1, n\2, if (gcd(k, n)==1, n%k)); \\ Michel Marcus, Jan 18 2021

Python

from math import gcd
def A340740(n): return sum(n % k for k in range(1,n//2+1) if gcd(k,n) == 1) # Chai Wah Wu, Mar 18 2021

Formula

a(n) = Sum_{k=1..floor((n-1)/2)} (n mod k) * floor(1/gcd(n,k)). - Wesley Ivan Hurt, Jan 18 2021