A300168 -id:A300168 - cp's OEIS frontend

A300162 Numbers of the form n^2 + 1 that can be expressed as j^2 + k^2, j > k > 1, in more ways than any smaller number of this form.

65, 325, 2210, 17425, 58565, 71825, 986050, 2458625, 15657850, 27636050, 205005125, 343842850, 1930020625, 4401924410, 5279766245, 26189596225, 109020153125, 1997821114250, 3001878503650, 5283874574225, 10991001911825, 91178970317825, 253647605037125

Offset: 1

Hugo Pfoertner, Feb 27 2018

nonn

			See A300161.

Giovanni Resta, Table of n, a(n) for n = 1..31

Cf. A052199, A071383, A299707, A299708, A300161, A300163, A300167, A300168.

a(17) from Hugo Pfoertner, Mar 08 2018
a(18)-a(21) from Robert Price, Mar 10 2018
a(22)-a(23) from Giovanni Resta, Mar 13 2018

A300163 Records in the number of ways to express a number of the form n^2 + 1 as j^2 + k^2 with j > k > 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 8, 11, 15, 17, 23, 31, 35, 39, 47, 63, 71, 95, 127, 143, 161, 191, 215, 255, 287, 319, 383, 575, 639, 767, 959, 1151

Offset: 1

Hugo Pfoertner, Feb 27 2018

			a(6) = 8 because A300162(6) = A300161(6)^2 + 1 = 71825 is the smallest number expressible in 8 ways: 71825 = 265^2 + 40^2 = 260^2 + 65^2 = 257^2 + 76^2 = 247^2 + 104^2 = 236^2 + 127^2 = 215^2 + 160^2 = 208^2 + 169^2 = 191^2 + 188^2.

Cf. A060306, A071385, A299707, A299708, A300161, A300162, A300168.

a(17) from Hugo Pfoertner, Mar 08 2018
a(18)-a(21) from Robert Price, Mar 10 2018
a(22)-a(31) from Giovanni Resta, Mar 13 2018

A300165 Numbers m such that m^2 + 1 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

47, 72, 73, 83, 98, 112, 122, 123, 128, 132, 133, 138, 142, 148, 157, 162, 172, 173, 174, 177, 183, 187, 191, 192, 200, 203, 208, 212, 213, 216, 217, 228, 233, 237, 242, 252, 253, 255, 265, 268, 273, 278, 288, 293, 294, 302, 307, 313, 317, 319

Offset: 1

Hugo Pfoertner, Feb 27 2018

nonn

The sequence differs from A299707 by the gcd condition, which excludes representations like 18^2 + 1 = 15^2 + 10^2, 32^2 + 1 = 25^2 + 20^2, 38^2 + 1 = 34^2 + 17^2.

			a(1) = 47 because its 3 representations satisfy the conditions j > k > 1 and gcd(j,k) = 1: 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, A299707, A300166, A300167, A300168.

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

A300162 Numbers of the form n^2 + 1 that can be expressed as j^2 + k^2, j > k > 1, in more ways than any smaller number of this form.

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A300163 Records in the number of ways to express a number of the form n^2 + 1 as j^2 + k^2 with j > k > 1.

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A300165 Numbers m such that m^2 + 1 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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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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