A362933 a(n) = (A132188(n) - A362931(n))/2.

0, 0, 0, 1, 1, -1, -1, 0, 2, 2, 2, 1, 1, 1, -1, 2, 2, 2, 2, 1, 1, 1, 1, 0, 4, 4, 6, 5, 5, 1, 1, 4, 4, 4, 2, 5, 5, 5, 5, 4, 4, 0, 0, 1, -1, -1, -1, 0, 6, 10, 10, 11, 11, 11, 11, 10, 10, 10, 10, 5, 5, 5, 5, 12, 12, 8, 8, 9, 9, 7, 7, 8, 8, 8, 10, 11, 9, 7, 7, 8

Offset: 1

Author

N. J. A. Sloane, Aug 28 2023

Keywords

• sign
• look

Comments

a(n) = one-half of (number of pairs (i,j) in [1..n] X [1..n] with integral geometric mean sqrt(i*j)) - (number of pairs (i,j) in [1..n] X [1..n] with integral harmonic mean 2*i*j/(i+j)).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..65536

Crossrefs

Cf. A132188, A362931, A362932.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 0, add(
      `if`(irem(2*i*n, i+n)=0, -1, 0)+
      `if`(issqr(i*n), 1, 0), i=1..n-1)+a(n-1))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Aug 28 2023

Python

from sympy.ntheory.primetest import is_square
def A362933(n): return sum((1 if T else -1) for x in range(1,n+1) for y in range(1,x) if (T:=is_square(x*y))^(not (x*y<<1)%(x+y))) # Chai Wah Wu, Aug 29 2023

Formula

a(n) = A362932(n)/2.