A063664 Numbers whose reciprocal is the sum of two reciprocals of squares.

2, 8, 18, 20, 32, 50, 72, 80, 90, 98, 128, 144, 162, 180, 200, 242, 272, 288, 320, 338, 360, 392, 450, 468, 500, 512, 576, 578, 648, 650, 720, 722, 800, 810, 882, 968, 980, 1058, 1088, 1152, 1250, 1280, 1296, 1332, 1352, 1440, 1458, 1568, 1620, 1682, 1800

Offset: 1

Author

Henry Bottomley, Jul 28 2001

Keywords

• nonn

Comments

These are numbers which can be written either as b^2*c^2*(b^2+c^2)*d^2 or if (b^2+c^2) is a square then as b^2*c^2*d^2, since 1/(b*(b^2+c^2)*d)^2+1/(c*(b^2+c^2)*d)^2 =1/(b^2*c^2*(b^2+c^2)*d^2) and 1/(b*sqrt(b^2+c^2)*d)^2+1/(c*sqrt(b^2+c^2)*d)^2 = 1/(b^2*c^2*d^2).

Examples

			98 is in the sequence since 1/98=1/10^2+1/70^2 (also 1/98=1/14^2+1/14^2).
		

Crossrefs

Either products of terms in A063663 and A000290, or squares of A008594.

Cf. A001481, A000404, A063665, A063669.

Programs

Python

from fractions import Fraction
def aupto(lim):
  sqr_recips = [Fraction(1, i*i) for i in range(1, lim+2)]
  ssr = set(f + g for i, f in enumerate(sqr_recips) for g in sqr_recips[i:])
  representable = [f.denominator for f in ssr if f.numerator == 1]
  return sorted(r for r in representable if r <= lim)
print(aupto(1800)) # Michael S. Branicky, Feb 08 2021

Extensions

Offset changed to 1 by Derek Orr, Jun 23 2015