A063665 -id:A063665 - cp's OEIS frontend

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

Henry Bottomley, Jul 28 2001

nonn

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).

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

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

Cf. A001481, A000404, A063665, A063669.

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

Offset changed to 1 by Derek Orr, Jun 23 2015

A063669 Hypotenuses of reciprocal Pythagorean triangles: number of solutions to 1/(12n)^2 = 1/b^2 + 1/c^2 [with b >= c > 0]; also number of values of A020885 (with repetitions) which divide n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 2, 1, 4, 1, 1, 1, 1, 4, 1, 1, 1, 1, 3, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 1, 2, 1, 1, 1, 1, 6, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 1, 1, 5

Offset: 1

Henry Bottomley, Jul 28 2001

nonn

Primitive reciprocal Pythagorean triangles 1/a^2 = 1/b^2 + 1/c^2 have a=fg, b=ef, c=eg where e^2 = f^2 + g^2; i.e., e,f,g represent the sides of primitive Pythagorean triangles. But the product of the two legs of primitive Pythagorean triangles are multiples of 12 and so the reciprocal of hypotenuses of reciprocal Pythagorean triangles are always multiples of 12 (A008594).

			a(1)=1 since 1/(12*1)^2 = 1/12^2 = 1/15^2 + 1/20^2;
a(70)=6 since 1/(12*70)^2 = 1/840^2 = 1/875^2 + 1/3000^2 = 1/888^2 + 1/2590^2 = 1/910^2 + 1/2184^2 = 1/952^2 + 1/1785^2 = 1/1050^2 + 1/1400^2 = 1/1160^2 + 1/1218^2.
Looking at A020885, 1 is divisible by 1, while 70 is divisible by 1, 5, 10, 14, 35 and again 35.

Cf. A046080, A063664, A063665, A063014.

cp's OEIS Frontend

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

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