cp's OEIS Frontend

A060992 a(n) = Sum_{gcd(i,j) | 0 < i <= j < n and i+j = n}.

%I A060992 #32 Jun 04 2025 00:23:29
%S A060992 0,1,1,3,2,6,3,8,6,11,5,17,6,16,15,20,8,27,9,31,22,26,11,44,20,31,27,
%T A060992 45,14,60,15,48,36,41,41,75,18,46,43,80,20,87,21,73,72,56,23,108,42,
%U A060992 85,57,87,26,108,67,116,64,71,29,165,30,76
%N A060992 a(n) = Sum_{gcd(i,j) | 0 < i <= j < n and i+j = n}.
%H A060992 Robert Israel, <a href="/A060992/b060992.txt">Table of n, a(n) for n = 1..10000</a>
%F A060992 a(n) = Sum_{d divides n} floor(d/2)*phi(n/d). a(p) = (p-1)/2 for an odd prime p. - _Vladeta Jovovic_, Dec 21 2004
%F A060992 a(n) = Sum_{i=1..floor(n/2)} gcd(n-i,i). - _Wesley Ivan Hurt_, Nov 12 2017
%F A060992 G.f.: Sum_{k>=1} phi(k)*x^(2*k)/((1 + x^k)*(1 - x^k)^2). - _Ilya Gutkovskiy_, Oct 24 2018
%F A060992 a(n) = (A018804(n) - A109043(n))/2. - _Ridouane Oudra_, May 31 2025
%e A060992 a(12) = gcd(1,11) + gcd(2,10) + gcd(3,9) + gcd(4,8) + gcd(5,7) + gcd(6,6) = 1 + 2 + 3 + 4 + 1 + 6 = 17;
%e A060992 a(13) = gcd(1,12) + gcd(2,11) + gcd(3,10) + gcd(4,9) + gcd(5,8) + gcd(6,7) = 1 + 1 + 1 + 1 + 1 + 1 = 6.
%p A060992 N:= 200: # to get a(1)..a(N)
%p A060992 A:= Vector(N):
%p A060992 for d from 1 to N do
%p A060992   c:= floor(d/2);
%p A060992   for n from d to N by d do
%p A060992     A[n]:= A[n]+c*numtheory:-phi(n/d)
%p A060992   od
%p A060992 od:
%p A060992 seq(A[i],i=1..N); # _Robert Israel_, May 11 2018
%t A060992 Table[Sum[GCD[n - i, i], {i, Floor[n/2]}], {n, 100}] (* _Wesley Ivan Hurt_, Nov 12 2017 *)
%o A060992 (PARI) a(n) = sumdiv(n, d, (d\2)*eulerphi(n/d)); \\ _Michel Marcus_, May 11 2018
%Y A060992 Cf. A234307, A018804, A109043.
%K A060992 nonn,easy,look
%O A060992 1,4
%A A060992 _Reinhard Zumkeller_, Feb 14 2002