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

1,1

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).

Links

Table of n, a(n) for n=1..51.

Example

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

Prog

(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

Crossrefs

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

Cf. A001481, A000404, A063665, A063669.

Sequence in context: A051248 A228615 A267823 * A094147 A117612 A320662

Adjacent sequences: A063661 A063662 A063663 * A063665 A063666 A063667

Keyword

nonn

Author

Henry Bottomley, Jul 28 2001

Extensions

Offset changed to 1 by Derek Orr, Jun 23 2015

Status

approved