A362932 a(n) = A132188(n) - A362931(n).

0, 0, 0, 2, 2, -2, -2, 0, 4, 4, 4, 2, 2, 2, -2, 4, 4, 4, 4, 2, 2, 2, 2, 0, 8, 8, 12, 10, 10, 2, 2, 8, 8, 8, 4, 10, 10, 10, 10, 8, 8, 0, 0, 2, -2, -2, -2, 0, 12, 20, 20, 22, 22, 22, 22, 20, 20, 20, 20, 10, 10, 10, 10, 24, 24, 16, 16, 18, 18, 14, 14, 16, 16, 16, 20, 22, 18, 14, 14, 16

Offset: 1

Author

N. J. A. Sloane, Aug 28 2023

Keywords

• sign
• look

Comments

a(n) = (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..10000

Crossrefs

Cf. A132188, A362931.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 0, 2*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

Mathematica

Block[{c, q}, c[n_] := c[n] = Flatten[Table[w^2 - x*y, {w, n}, {x, n}, {y, n}]]; q[n_] := q[n] = Flatten[Table[(2*i*j)/(i + j), {i, n}, {j, n}]]; Table[Count[c[n], 0] - Count[q[n], ?IntegerQ], {n, 80}] ] (* _Michael De Vlieger, Aug 28 2023 *)

Python

from sympy.ntheory.primetest import is_square
def A362932(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)))<<1 # Chai Wah Wu, Aug 28 2023