A057789 a(n) = Sum_{k = 1..n, gcd(k,n)=1} k*(n-k).

0, 1, 4, 6, 20, 10, 56, 44, 84, 60, 220, 92, 364, 182, 280, 344, 816, 318, 1140, 520, 840, 770, 2024, 760, 2100, 1300, 2196, 1540, 4060, 1240, 4960, 2736, 3520, 2992, 4760, 2580, 8436, 4218, 5928, 4240, 11480, 3612, 13244, 6380, 8040, 7590, 17296, 6128

Offset: 1

Author

Leroy Quet, Nov 04 2000

Keywords

• nonn
• look

Comments

Equal to convolution sum over positive integers, k, where k<=n and gcd(k,n)=1, except in first term, where the convolution sum is 1 instead of 0.

Examples

			Since 1, 3, 5 and 7 are relatively prime to 8 and are <= 8, a(8) = 1*(8-1) +3*(8-3) +5*(8-5) +7*(8-7) = 44.
		

Links

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

Crossrefs

Cf. A000292, A023896, A053818, A053819, A002618, A053191, A023900.

Programs

Maple

f:= proc(n) local i;
  2*add(`if`(igcd(i,n)=1, i*(n-i),0),i=1..n/2)
end proc:
f(2):= 1:
map(f, [$1..100]); # Robert Israel, Sep 29 2019

Mathematica

a[n_] := 2 Sum[Boole[CoprimeQ[k, n]] k (n - k), {k, 1, n/2}];
a[2] = 1;
Array[a, 100] (* Jean-François Alcover, Aug 16 2020, after Maple *)

PARI

a(n) = sum(k=1, n, if (gcd(n,k)==1, k*(n-k))); \\ Michel Marcus, Sep 29 2019

Formula

From Robert Israel, Sep 29 2019: (Start)

If n is prime, a(n) = A000292(n-1).

If n/2 is an odd prime, a(n) = A000292(n-2)/2.

If n/3 is a prime other than 3, a(n) = A000292(n-3)*2*n/(3*(n-2)). (End)

From Ridouane Oudra, Mar 21 2024: (Start)

a(n) = n*A023896(n) - A053818(n) ;

a(n) = (2/3)*(n*A023896(n) - A053819(n)/n) ;

a(n) = (n/6)*(A002618(n) - A023900(n)) ;

a(n) = (1/6)*(A053191(n) - n*A023900(n)). (End)

Sum_{k=1..n} a(k) ~ n^4 / (4*Pi^2). - Amiram Eldar, Apr 11 2024