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

%I A057789 #30 Apr 11 2024 04:05:49
%S A057789 0,1,4,6,20,10,56,44,84,60,220,92,364,182,280,344,816,318,1140,520,
%T A057789 840,770,2024,760,2100,1300,2196,1540,4060,1240,4960,2736,3520,2992,
%U A057789 4760,2580,8436,4218,5928,4240,11480,3612,13244,6380,8040,7590,17296,6128
%N A057789 a(n) = Sum_{k = 1..n, gcd(k,n)=1} k*(n-k).
%C A057789 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.
%H A057789 Robert Israel, <a href="/A057789/b057789.txt">Table of n, a(n) for n = 1..10000</a>
%F A057789 From _Robert Israel_, Sep 29 2019: (Start)
%F A057789 If n is prime, a(n) = A000292(n-1).
%F A057789 If n/2 is an odd prime, a(n) =  A000292(n-2)/2.
%F A057789 If n/3 is a prime other than 3, a(n) = A000292(n-3)*2*n/(3*(n-2)). (End)
%F A057789 From _Ridouane Oudra_, Mar 21 2024: (Start)
%F A057789 a(n) = n*A023896(n) - A053818(n) ;
%F A057789 a(n) = (2/3)*(n*A023896(n) - A053819(n)/n) ;
%F A057789 a(n) = (n/6)*(A002618(n) - A023900(n)) ;
%F A057789 a(n) = (1/6)*(A053191(n) - n*A023900(n)). (End)
%F A057789 Sum_{k=1..n} a(k) ~ n^4 / (4*Pi^2). - _Amiram Eldar_, Apr 11 2024
%e A057789 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.
%p A057789 f:= proc(n) local i;
%p A057789   2*add(`if`(igcd(i,n)=1, i*(n-i),0),i=1..n/2)
%p A057789 end proc:
%p A057789 f(2):= 1:
%p A057789 map(f, [$1..100]); # _Robert Israel_, Sep 29 2019
%t A057789 a[n_] := 2 Sum[Boole[CoprimeQ[k, n]] k (n - k), {k, 1, n/2}];
%t A057789 a[2] = 1;
%t A057789 Array[a, 100] (* _Jean-François Alcover_, Aug 16 2020, after Maple *)
%o A057789 (PARI) a(n) = sum(k=1, n, if (gcd(n,k)==1, k*(n-k))); \\ _Michel Marcus_, Sep 29 2019
%Y A057789 Cf. A000292, A023896, A053818, A053819, A002618, A053191, A023900.
%K A057789 nonn,look
%O A057789 1,3
%A A057789 _Leroy Quet_, Nov 04 2000