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A063664
Numbers whose reciprocal is the sum of two reciprocals of squares.
5
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).
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.
Sequence in context: A051248 A228615 A267823 * A094147 A117612 A320662
KEYWORD
nonn
AUTHOR
Henry Bottomley, Jul 28 2001
EXTENSIONS
Offset changed to 1 by Derek Orr, Jun 23 2015
STATUS
approved