A234307 -id:A234307 - cp's OEIS frontend

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

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, 45, 14, 60, 15, 48, 36, 41, 41, 75, 18, 46, 43, 80, 20, 87, 21, 73, 72, 56, 23, 108, 42, 85, 57, 87, 26, 108, 67, 116, 64, 71, 29, 165, 30, 76

Offset: 1

Reinhard Zumkeller, Feb 14 2002

			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;
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.

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

Cf. A234307, A018804, A109043.

N:= 200: # to get a(1)..a(N)
A:= Vector(N):
for d from 1 to N do
  c:= floor(d/2);
  for n from d to N by d do
    A[n]:= A[n]+c*numtheory:-phi(n/d)
  od
od:
seq(A[i],i=1..N); # Robert Israel, May 11 2018

Table[Sum[GCD[n - i, i], {i, Floor[n/2]}], {n, 100}] (* Wesley Ivan Hurt, Nov 12 2017 *)

a(n) = sumdiv(n, d, (d\2)*eulerphi(n/d)); \\ Michel Marcus, May 11 2018

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
a(n) = Sum_{i=1..floor(n/2)} gcd(n-i,i). - Wesley Ivan Hurt, Nov 12 2017
G.f.: Sum_{k>=1} phi(k)*x^(2*k)/((1 + x^k)*(1 - x^k)^2). - Ilya Gutkovskiy, Oct 24 2018
a(n) = (A018804(n) - A109043(n))/2. - Ridouane Oudra, May 31 2025

A280448 Sum of the GCDs of the smaller and larger parts of the partitions of 2n into two squarefree parts.

Original entry on oeis.org

1, 3, 4, 4, 6, 10, 9, 7, 6, 20, 15, 11, 17, 28, 19, 11, 23, 23, 25, 27, 36, 48, 30, 24, 12, 55, 16, 35, 39, 56, 41, 20, 55, 73, 55, 44, 50, 81, 65, 39, 53, 96, 56, 71, 33, 97, 63, 40, 29, 53, 88, 83, 71, 63, 91, 68, 98, 126, 78, 87, 80, 134, 65, 40, 107, 147, 89, 107, 119

Offset: 1

Wesley Ivan Hurt, Jan 03 2017

Index entries for sequences related to partitions

Cf. A008683, A234307, A280226.

with(numtheory): A280448:=n->add(gcd(2*n-i, i)*mobius(i)^2*mobius(2*n-i)^2, i=1..n): seq(A280448(n), n=1..100);

Table[Sum[GCD[k, 2*n - k]*MoebiusMu[k]^2 * MoebiusMu[2*n - k]^2, {k, 1,
n}], {n, 1, 50}] (* G. C. Greubel, Jan 05 2017 *)

for(n=1,50, print1(sum(k=1,n, gcd(k,2*n-k) * (moebius(k))^2 *(moebius(2*n-k))^2), ", ")) \\ G. C. Greubel, Jan 05 2017

a(n) = Sum_{i=1..n} gcd(i,2n-i) * mu(i)^2 * mu(2n-i)^2, where mu is the Möbius function (A008683).

cp's OEIS Frontend

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

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