A300165 -id:A300165 - cp's OEIS frontend

A299707 Numbers m such that m^2 + 1 can be expressed in more than one way as j^2 + k^2 with j > k > 1.

18, 32, 38, 43, 47, 57, 68, 70, 72, 73, 82, 83, 93, 98, 99, 107, 112, 117, 118, 122, 123, 128, 132, 133, 138, 142, 143, 148, 157, 162, 168, 172, 173, 174, 177, 182, 183, 187, 191, 192, 193, 200, 203, 207, 208, 212, 213, 216, 217, 218, 228, 232, 233, 237, 242, 243, 251, 252

Offset: 1

Hugo Pfoertner, Feb 27 2018

nonn

			a(1) = 18: 18^2 + 1 = 325 = 17^2 + 6^2 = 15^2 + 10^2,
a(2) = 32: 32^2 + 1 = 1025 = 31^2 + 8^2 = 25^2 + 20^2,
a(5) = 47: 47^2 + 1 = 2210 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A050796, A299708, A300161, A300162, A300163, A300164, A300165, A300166.

A299708 Numbers of the form m^2 + 1 that can be expressed in more than one way as j^2 + k^2 with j > k > 1.

Original entry on oeis.org

325, 1025, 1445, 1850, 2210, 3250, 4625, 4901, 5185, 5330, 6725, 6890, 8650, 9605, 9802, 11450, 12545, 13690, 13925, 14885, 15130, 16385, 17425, 17690, 19045, 20165, 20450, 21905, 24650, 26245, 28225, 29585, 29930, 30277, 31330, 33125, 33490

Offset: 1

Hugo Pfoertner, Feb 27 2018

nonn

			a(1) = 325 = A299707(1)^2 + 1 = 18^2 + 1 is expressible in two ways:
  325 = 17^2 + 6^2 = 15^2 + 10^2.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A050796, A299707, A300161, A300162, A300163, A300164, A300165, A300166.

A300166 Numbers of the form m^2 + 1 that can be expressed in more than one way as j^2 + k^2 with j > k > 1 and gcd(j,k) = 1.

Original entry on oeis.org

2210, 5185, 5330, 6890, 9605, 12545, 14885, 15130, 16385, 17425, 17690, 19045, 20165, 21905, 24650, 26245, 29585, 29930, 30277, 31330, 33490, 34970, 36482, 36865, 40001, 41210, 43265, 44945, 45370, 46657, 47090, 51985, 54290, 56170, 58565, 63505

Offset: 1

Hugo Pfoertner, Feb 27 2018

nonn

The sequence differs from A299708 by the gcd condition, which excludes representations like 325 = 18^2 + 1^2 = 15^2 + 10^2, 1025 = 32^2 + 1 = 25^2 + 20^2, 1445 = 38^2 + 1 = 34^2 + 17^2.

			a(1) = 2210 because its 3 representations satisfy the conditions j > k > 1 and gcd(j,k) = 1: 2210 = 47^2 + 1 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A050796, A299707, A299708, A300165, A300167, A300168.

A300167 Numbers n such that n^2+1 can be expressed as j^2+k^2, j>k>1, gcd(j,k)=1, in more ways than for any smaller n.

Original entry on oeis.org

8, 47, 242, 2163, 21042, 72662, 1555572, 16485763, 169053487, 2017326722

Offset: 1

Hugo Pfoertner, Feb 27 2018

			a(1)= 8 because 8^2 + 1 = A300168(1) = 65 = 7^2 + 4^2.
a(2) = 47 because it is the smallest n leading to more than 1 way of expressing n^2+1 : 47^2 + 1 = 2010 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2.
a(3) = 242 because 242^2 + 1 = 58565 is the smallest number that can be expressed in more than 3 ways:
  58565 = 241^2 + 22^2 = 239^2 + 38^2 = 223^2 + 94^2 = 214^2 + 113^2 = 209^2 + 122^2 = 206^2 + 127^2 = 193^2 + 146^2.

Cf. A300161, A300165, A300168.

a(7) from Robert Price, Mar 11 2018
a(7) corrected, a(8)-a(9) added by Ray Chandler, Dec 23 2019
a(10) added by Ray Chandler, Dec 31 2019

cp's OEIS Frontend

A299707 Numbers m such that m^2 + 1 can be expressed in more than one way as j^2 + k^2 with j > k > 1.

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A299708 Numbers of the form m^2 + 1 that can be expressed in more than one way as j^2 + k^2 with j > k > 1.

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A300166 Numbers of the form m^2 + 1 that can be expressed in more than one way as j^2 + k^2 with j > k > 1 and gcd(j,k) = 1.

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A300167 Numbers n such that n^2+1 can be expressed as j^2+k^2, j>k>1, gcd(j,k)=1, in more ways than for any smaller n.

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