A369333 Positive integers m such that there exist distinct pairs (a,b) and (c,d) with a <= b, c <= d, and m = a*b*(c+d) = (a+b)*c*d.

72, 144, 168, 360, 450, 480, 576, 864, 990, 1152, 1200, 1344, 1404, 1568, 1600, 1800, 1944, 2040, 2160, 2520, 2646, 2880, 3150, 3360, 3600, 3780, 3840, 3888, 4050, 4536, 4608, 4800, 5184, 5400, 5520, 5880, 5940, 6720, 6912, 7056, 7200, 7350, 7800, 7920, 7938, 8550, 8640, 8694, 8712, 8976, 9000, 9216, 9408, 9450, 9504, 9600, 9720, 10416, 10752, 11232, 11550, 11760

Offset

1,1

Comments

Such numbers m correspond to pairs of equal Egyptian fractions of length 2, since a*b*(c+d) = (a+b)*c*d is equivalent to 1/a + 1/b = 1/c + 1/d.

Numbers of the form t * k^3, where t is a term of A369334 and k is a positive integer.

If m belongs to this sequence, then so does m*k^3 for any positive integer k.

Links

Table of n, a(n) for n=1..62.

Example

72 is a term since 72 = 3*3*(2+6) = (3+3)*2*6.

Prog

(PARI) { find_abcd(m) = my(r); fordiv(m, a, if(2*a*a>m, break); fordiv(m\a, b, if(4*a^2*b^2 > m*(a+b), break); if(b<a || m%(a+b), next); if(a*b*(a+b)==m, next); r=select(x->denominator(x)==1, nfroots(, x^2 - m\a\b*x + m\(a+b))); if(#r==1, r=[r[1], r[1]]); if(#r==2, return([a, b, r[1], r[2]])); )); []; }

Crossrefs

Cf. A088915, A369334, A371721.

Sequence in context: A345541 A345794 A157336 * A369334 A060661 A050495

Adjacent sequences: A369330 A369331 A369332 * A369334 A369335 A369336

Keyword

nonn

Author

Max Alekseyev, Jan 20 2024

Status

approved