A299708 -id:A299708 - 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.

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.

Original entry on oeis.org

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

A300161 Numbers n such that n^2 + 1 can be expressed as j^2 + k^2, j > k > 1, in more ways than for any smaller n.

Original entry on oeis.org

8, 18, 47, 132, 242, 268, 993, 1568, 3957, 5257, 14318, 18543, 43932, 66347, 72662, 161832, 330182, 1413443, 1732593, 2298668, 3315268, 9548768, 15926318, 24310918, 27028568, 51853693, 162166243, 420024818, 472936732, 599832943, 1892369318

Offset: 1

Hugo Pfoertner, Feb 27 2018

			a(1) = 8: 8^2 + 1 = 65 = 7^2 + 4^2,
a(2) = 18: 18^2 + 1 = 325 = 17^2 + 6^2 = 15^2 + 10^2,
a(3) = 47: 47^2 + 1 = 2210 = 43^2 + 19^2 = 41^2 + 23^2 = 37^2 + 29^2,
a(4) = 132: 132^2 + 1 = 17425 = 129^2 + 28^2 = 127^2 + 36^2 = 120^2 + 55^2 = 116^2 + 63^2 = 105^2 + 80^2.

Cf. A050796, A299707, A299708, A300162, A300163.

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

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

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.

A300164 Numbers of the form n^2+1 not expressible as j^2+k^2 with j>k>1.

Original entry on oeis.org

2, 5, 10, 17, 26, 37, 50, 82, 101, 122, 197, 226, 257, 362, 401, 577, 626, 677, 842, 1226, 1297, 1522, 1601, 1682, 2026, 2402, 2602, 2917, 3137, 3482, 3722, 4226, 4357, 4762, 5042, 5477, 6242, 7057, 7226, 8101, 8837, 9026, 10202, 12101, 13457, 14401

Offset: 1

Hugo Pfoertner, Feb 27 2018

nonn

			The first numbers of the form n^2 + 1 not in the sequence are:
  65 = 8^2 + 1 because it can be expressed as 65 = 7^2 + 4^2,
  145 = 12^2 + 1 = 9^2 + 8^2,
  170 = 13^2 + 1 = 11^2 + 7^2.

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

Cf. A050796, A065876, A071557, A299708.

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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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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A300161 Numbers n such that n^2 + 1 can be expressed as j^2 + k^2, j > k > 1, in more ways than for any smaller n.

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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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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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A300164 Numbers of the form n^2+1 not expressible as j^2+k^2 with j>k>1.

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