A375512 - OEIS

A375512

a(n) is the number of distinct compositions of four positive integers x, u, v, y (x < u <= v < y) for which x + u + v + y = n and u*v = x*y.

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 0, 2, 3, 0, 0, 4, 3, 0, 4, 3, 0, 6, 0, 3, 5, 0, 6, 9, 0, 0, 6, 8, 0, 9, 0, 5, 13, 0, 0, 13, 6, 7, 8, 6, 0, 11, 10, 12, 9, 0, 0, 23, 0, 0, 19, 10, 12, 15, 0, 8, 11, 18, 0, 27, 0, 0, 23, 9, 15, 18, 0, 25, 19

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OFFSET

0,16

COMMENTS

(bin(4,0) + bin(4,2) + bin(4,4))*a(n) = 8*a(n) is the number of distinct compositions of four integers x, u, v, y (abs(x) < abs(u) <= abs(v) < abs(y)) for which abs(x) + abs(u) + abs(v) + abs(y) = n and u*v = x*y.

a(n) is also the number of 2X2 matrices having the determinant 0 whose elements [x,u;v,y] are positive integers with x < u <= v < y and x + u + v + y = n.

a(n) is also the number of distinct linear 2X2 equation systems that do not have exactly one solution and whose coefficients [x,u;v,y] are positive integers with x < u <= v < y and x + u + v + y = n.

LINKS

Felix Huber, Table of n, a(n) for n = 0..1000

Felix Huber, Maple codes

FORMULA

Conjecture: a(p) = 0 for primes p.

From Robert Israel, Aug 23 2024: (Start)

The conjecture is true, in fact for any x,y,u,v as in the definition, n has proper divisor gcd(x,u) + gcd(v,y).

Proof: Suppose x,y,u,v are positive integers with x + y + u + v = n and x*y = u*v = m. Let g = gcd(x,u). Then x = g*X and u = g*U where X and U are coprime. Since X*y = U*v = m/g, we must have y = h*U and v = h*X where h = gcd(v,y). Then n = g*X + h*U + g*U + h*X = (g+h)*(U+X).

(End)

EXAMPLE

a(9) = 1 because only (1, 2, 2, 4) satisfies the conditions: 1 + 2 + 2 + 4 = 9 and 2*2 = 1*4.

a(24) = 4 because (1, 2, 7, 14), (1, 3, 5, 15), (2, 4, 6, 12), (3, 5, 6, 10) satisfy the conditions: 1 + 2 + 7 + 14 = 24 and 2*7 = 1*14, 1 + 3 + 5 + 15 = 24 and 3*5 = 1*15, 2 + 4 + 6 + 12 = 24 and 4*6 = 2*12, 3 + 5 + 6 + 10 = 24 and 5*6 = 3*10.