A300163 -id:A300163 - cp's OEIS frontend

A071385 Number of points (i,j) on the circumference of a circle around (0,0) with squared radius A071383(n).

1, 4, 8, 12, 16, 24, 32, 36, 48, 64, 72, 80, 96, 128, 144, 160, 192, 256, 288, 320, 384, 512, 576, 640, 768, 864, 1024, 1152, 1280, 1536, 1728, 2048, 2304, 2560, 3072, 3456, 3840, 4096, 4608, 5120, 6144, 6912, 7680, 8192, 9216, 10240, 11520, 12288, 13824, 15360

Offset: 1

Hugo Pfoertner, May 23 2002

nonn

			Circles with radius 1 and 2 have 4 lattice points on their circumference, so a(1)=4. A circle with radius sqrt(5) passes through 8 lattice points of the shape (2,1), so a(2)=8. A circle with radius 5 passes through 4 lattice points of shape (5,0) and through 8 points of shape (4,3), so a(3)=4+8=12
A071383(11) = 5^2 * 13^2 * 17^1 = 71825. Therefore A071385(11) = 4*(2+1)*(2+1)*(1+1) = 72.

Ray Chandler, Table of n, a(n) for n = 1..425
Hugo Pfoertner, Construction of A071383, A071384, A071385

Records in A004018. Cf. A071383, A071384, A002144, A002654, A300163, A344470.

my(v=list(10^15), rec=0); print1(1, ", "); for(n=1, #v, if(numdiv(v[n])>rec, rec=numdiv(v[n]); print1(4*rec, ", "))) \\ Jianing Song, May 20 2021, see program for A054994

a(n) = 4 * Product_{k=1..klim} (e_k + 1), where klim and e_1 >= e_2 >= ... >= e_klim > 0 are known from A071383(n) = Product_{k=1..klim} p_k^e_k, with p_k = k-th prime of the form 4i+1. (J. H. Conway)
a(n) = 4*A000005(A071383(n)) for n > 1.
a(n) = A004018(A071383(n)).
a(n) = A002654(A071383(n)) for n > 1. - Jianing Song, May 20 2021
a(n) = 4*A344470(n-1) for n > 1. - Hugo Pfoertner, Sep 04 2022

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.

Original entry on oeis.org

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.

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

cp's OEIS Frontend

A071385 Number of points (i,j) on the circumference of a circle around (0,0) with squared radius A071383(n).

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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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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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