A257408 -id:A257408 - cp's OEIS frontend

A257409 Values of n such that there are exactly 2 solutions to x^2 - y^2 = n, with x > y >= 0.

9, 15, 16, 21, 24, 25, 27, 32, 33, 35, 36, 39, 40, 49, 51, 55, 56, 57, 60, 65, 69, 77, 84, 85, 87, 88, 91, 93, 95, 100, 104, 108, 111, 115, 119, 121, 123, 125, 129, 132, 133, 136, 140, 141, 143, 145, 152, 155, 156, 159, 161, 169, 177, 183, 184, 185, 187, 196

Offset: 1

Colin Barker, Apr 22 2015

nonn

A subsequence of A058957. Terms in the latter but not here are 45, 48, 63, 64, 72, 75, 80, 81, 96, 99, ... - M. F. Hasler, Apr 22 2015

			9 is in the sequence because there are 2 solutions to x^2 - y^2 = 9, namely (x,y) = (3,0), (5,4).

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A257408, A257410-A257417.

Cf. A058957, A034178.

r[n_] := Reduce[x^2 - y^2 == n && x > y >= 0, {x, y}, Integers]; Reap[For[n = 1, n < 200, n++, rn = r[n]; If[rn[[0]] === Or && Length[rn] == 2, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Apr 22 2015 *)

is_A257409(n)={A034178(n)==2} \\ M. F. Hasler, Apr 22 2015

A257411 Values of n such that there are exactly 4 solutions to x^2 - y^2 = n with x > y >= 0.

Original entry on oeis.org

96, 105, 120, 135, 160, 165, 168, 189, 195, 216, 224, 231, 255, 256, 264, 273, 280, 285, 297, 312, 345, 351, 352, 357, 375, 385, 399, 408, 416, 420, 429, 435, 440, 455, 456, 459, 465, 483, 512, 513, 520, 540, 544, 552, 555, 561, 595, 608, 609, 615, 616, 621

Offset: 1

Colin Barker, Apr 22 2015

nonn

			96 is in the sequence because there are 4 solutions to x^2 - y^2 = 96, namely (x,y) = (10,2), (11,5), (14,10), (25,23).

Colin Barker, Table of n, a(n) for n = 1..600

Cf. A257408, A257409, A257410, A257412, A257413, A257414, A257415, A257416, A257417.

Cf. A034178.

nn = 1000;
t = Table[0, {nn}];
Do[n = x^2 - y^2; If[n <= nn, t[[n]]++], {x, nn}, {y, 0, x - 1}];
Position[t, 4] // Flatten (* Jean-François Alcover, Jun 18 2020, after T. D. Noe in A034178 *)

is_A257411(n)={A034178(n)==4} \\ M. F. Hasler, Apr 22 2015

A257417 Values of n such that there are exactly 10 solutions to x^2 - y^2 = n with x > y >= 0.

Original entry on oeis.org

960, 1344, 1728, 2112, 2240, 2496, 2592, 2835, 3240, 3264, 3520, 3648, 4160, 4416, 4455, 4536, 4928, 5265, 5440, 5568, 5824, 5952, 6080, 6144, 6237, 6885, 7104, 7128, 7360, 7371, 7616, 7695, 7872, 8000, 8256, 8424, 8512, 9024, 9152, 9280, 9315, 9639, 9920

Offset: 1

Colin Barker, Apr 22 2015

nonn

			960 is in the sequence because there are 10 solutions to x^2 - y^2 = 960, namely (x,y) = (31,1), (32,8), (34,14), (38,22), (46,34), (53,43), (64,56), (83,77), (122,118), (241,239).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 200 terms from Colin Barker)

Cf. A257408, A257409, A257410, A257411, A257412, A257413, A257414, A257415, A257416.

Cf. A034178.

nn = 10000;
t = Table[0, {nn}];
Do[n = x^2 - y^2; If[n <= nn, t[[n]]++], {x, nn}, {y, 0, x - 1}];
Position[t, 10] // Flatten (* Jean-François Alcover, Jun 18 2020, after T. D. Noe in A034178 *)

is_A257417(n)={A034178(n)==10} \\ M. F. Hasler, Apr 22 2015

A257410 Values of n such that there are exactly 3 solutions to x^2 - y^2 = n with x > y >= 0.

Original entry on oeis.org

45, 48, 63, 64, 72, 75, 80, 81, 99, 112, 117, 128, 147, 153, 171, 175, 176, 180, 200, 207, 208, 243, 245, 252, 261, 272, 275, 279, 300, 304, 324, 325, 333, 363, 368, 369, 387, 392, 396, 423, 425, 464, 468, 475, 477, 496, 507, 531, 539, 549, 575, 588, 592

Offset: 1

Colin Barker, Apr 22 2015

nonn

			45 is in the sequence because there are 3 solutions to x^2 - y^2 = 45, namely (x,y) = (7,2),(9,6),(23,22).

Colin Barker, Table of n, a(n) for n = 1..500

Cf. A257408, A257409, A257411-A257417.

Cf. A034178.

r[n_] := Reduce[x^2 - y^2 == n && x > y >= 0, {x, y}, Integers]; Reap[For[n = 1, n < 600, n++, rn = r[n]; If[rn[[0]] === Or && Length[rn] == 3, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Apr 22 2015 *)

is_A257410(n)={A034178(n)==3} \\ M. F. Hasler, Apr 22 2015

A257412 Values of n such that there are exactly 5 solutions to x^2 - y^2 = n with x > y >= 0.

Original entry on oeis.org

144, 192, 225, 320, 400, 405, 441, 448, 567, 648, 704, 784, 832, 891, 900, 1024, 1053, 1088, 1089, 1216, 1225, 1377, 1472, 1521, 1539, 1620, 1764, 1856, 1863, 1875, 1936, 1984, 2048, 2268, 2349, 2368, 2511, 2601, 2624, 2704, 2752, 2997, 3008, 3025, 3249

Offset: 1

Colin Barker, Apr 22 2015

nonn

			144 is in the sequence because there are 5 solutions to x^2 - y^2 = 144, namely (x,y) = (12,0), (13,5), (15,9), (20,16), (37,35).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 150 terms from Colin Barker)

Cf. A257408, A257409, A257410, A257411, A257413, A257414, A257415, A257416, A257417.

Cf. A034178.

nn = 4000;
t = Table[0, {nn}];
Do[n = x^2 - y^2; If[n <= nn, t[[n]]++], {x, nn}, {y, 0, x - 1}];
Position[t, 5] // Flatten (* Jean-François Alcover, Jun 18 2020, after T. D. Noe in A034178 *)

is_A257412(n)={A034178(n)==5} \\ M. F. Hasler, Apr 22 2015

A257413 Values of n such that there are exactly 6 solutions to x^2 - y^2 = n with x > y >= 0.

Original entry on oeis.org

240, 288, 315, 336, 360, 384, 432, 495, 504, 525, 528, 560, 585, 600, 624, 640, 675, 693, 735, 765, 792, 800, 816, 819, 825, 855, 880, 896, 912, 936, 975, 1035, 1040, 1071, 1104, 1125, 1176, 1197, 1215, 1224, 1232, 1260, 1275, 1287, 1305, 1323, 1360, 1368

Offset: 1

Colin Barker, Apr 22 2015

nonn

			240 is in the sequence because there are 6 solutions to x^2 - y^2 = 240, namely (x,y) = (16,4), (17,7), (19,11), (23,17), (32,28), (61,59).

Colin Barker, Table of n, a(n) for n = 1..1100

Cf. A257408, A257409, A257410, A257411, A257412, A257414, A257415, A257416, A257417.

Cf. A034178.

nn = 2000;
t = Table[0, {nn}];
Do[n = x^2 - y^2; If[n <= nn, t[[n]]++], {x, nn}, {y, 0, x - 1}];
Position[t, 6] // Flatten (* Jean-François Alcover, Jun 18 2020, after T. D. Noe in A034178 *)

is_A257413(n)={A034178(n)==6} \\ M. F. Hasler, Apr 22 2015

A257414 Values of n such that there are exactly 7 solutions to x^2 - y^2 = n with x > y >= 0.

Original entry on oeis.org

768, 1280, 1792, 2816, 3328, 3645, 4352, 4864, 5103, 5832, 5888, 7424, 7936, 8019, 9472, 9477, 10496, 11008, 12032, 12393, 13568, 13851, 14580, 15104, 15616, 16384, 16767, 17152, 18176, 18688, 20224, 20412, 21141, 21248, 22599, 22784, 24832, 25856, 26368

Offset: 1

Colin Barker, Apr 22 2015

nonn

			768 is in the sequence because there are 7 solutions to x^2 - y^2 = 768, namely (x,y) = (28,4), (32,16), (38,26), (52,44), (67,61), (98,94), (193,191).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 120 terms from Colin Barker)

Cf. A257408, A257409, A257410, A257411, A257412, A257413, A257415, A257416, A257417.

Cf. A034178, A068314.

nn = 30000;
t = Table[0, {nn}];
Do[n = x^2 - y^2; If[n <= nn, t[[n]]++], {x, nn}, {y, 0, x - 1}];
Position[t, 7] // Flatten (* Jean-François Alcover, Jun 18 2020, after T. D. Noe in A034178 *)

is_A257414(n)={A034178(n)==7} \\ M. F. Hasler, Apr 22 2015

A257415 Values of n such that there are exactly 8 solutions to x^2 - y^2 = n with x > y >= 0.

Original entry on oeis.org

480, 576, 672, 840, 864, 945, 1056, 1080, 1120, 1155, 1248, 1296, 1320, 1365, 1485, 1512, 1536, 1560, 1600, 1632, 1755, 1760, 1785, 1824, 1848, 1995, 2025, 2040, 2079, 2080, 2145, 2184, 2208, 2280, 2295, 2376, 2415, 2457, 2464, 2560, 2565, 2625, 2720, 2760

Offset: 1

Colin Barker, Apr 22 2015

nonn

			480 is in the sequence because there are 8 solutions to x^2 - y^2 = 480, namely (x,y) = (22,2), (23,7), (26,14), (29,19), (34,26), (43,37), (62,58), (121,119).

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1600 terms from Colin Barker)

Cf. A257408, A257409, A257410, A257411, A257412, A257413, A257414, A257416, A257417.

Cf. A034178.

nn = 3000;
t = Table[0, {nn}];
Do[n = x^2 - y^2; If[n <= nn, t[[n]]++], {x, nn}, {y, 0, x - 1}];
Position[t, 8] // Flatten (* Jean-François Alcover, Jun 18 2020, after T. D. Noe in A034178 *)

is_A257415(n)={A034178(n)==8} \\ M. F. Hasler, Apr 22 2015

A257416 Values of n such that there are exactly 9 solutions to x^2 - y^2 = n with x > y >= 0.

Original entry on oeis.org

720, 1008, 1152, 1200, 1575, 1584, 1800, 1872, 2205, 2352, 2448, 2475, 2736, 2800, 2925, 3072, 3200, 3312, 3528, 3675, 3825, 3888, 3920, 4176, 4275, 4400, 4464, 4851, 5120, 5175, 5200, 5328, 5445, 5733, 5808, 5904, 6075, 6192, 6272, 6300, 6525, 6768, 6800

Offset: 1

Colin Barker, Apr 22 2015

nonn

Numbers of the following forms: p[1]*p[2]^2*p[3]^2, p[1]^2*p[2]^5, p[1]*p[2]^8, p[1]^17, 2^2*p[1]*p[2]^2*p[3]^2, 2^2*p[1]^2*p[2]^5, 2^3*p[1]^2*p[2]^2, 2^3*p[1]^8, 2^4*p[1]*p[2]^2, 2^4*p[1]^5, 2^7*p[1]^2, 2^10*p[1], 2^19, where p[i] are distinct odd primes. - Robert Israel, Jun 19 2018

			720 is in the sequence because there are 9 solutions to x^2 - y^2 = 720, namely (x,y) = (27,3), (28,8), (29,11), (36,24), (41,31), (49,41), (63,57), (92,88), (181,179).

Robert Israel, Table of n, a(n) for n = 1..10000 (first 260 terms from Colin Barker)

Cf. A257408, A257409, A257410, A257411, A257412, A257413, A257414, A257415, A257417.

Cf. A034178.

filter:= proc(n) local k;
k:= padic:-ordp(n,2);
(k = 0 and numtheory:-tau(n)=18) or (k-1)*numtheory:-tau(n/2^k)=18
end proc:
select(filter, [$1..10^4]); # Robert Israel, Jun 19 2018

nn = 6800;
t = Table[0, {nn}];
Do[n = x^2 - y^2; If[n <= nn, t[[n]]++], {x, nn}, {y, 0, x - 1}];
Position[t, 9] // Flatten (* Jean-François Alcover, Jun 18 2020, after T. D. Noe in A034178 *)

is_A257416(n)={A034178(n)==9} \\ M. F. Hasler, Apr 22 2015

A298865 The primes p and products 4*p in increasing order.

Original entry on oeis.org

2, 3, 5, 7, 8, 11, 12, 13, 17, 19, 20, 23, 28, 29, 31, 37, 41, 43, 44, 47, 52, 53, 59, 61, 67, 68, 71, 73, 76, 79, 83, 89, 92, 97, 101, 103, 107, 109, 113, 116, 124, 127, 131, 137, 139, 148, 149, 151, 157, 163, 164, 167, 172, 173, 179, 181, 188, 191, 193

Offset: 1

Clark Kimberling, Feb 13 2018

Conjecture: except for the first term, these are the nonsquares n for which there is a unique pair (x,y) such that x^2 - y^2 = n and x > y >= 0; see A257408.

Clark Kimberling, Table of n, a(n) for n = 1..1000

Cf. A000040, A298866, A298867.

z = 10000; u = Prime[Range[z]]; w = Take[Union[u, 4 u], z]; (* A298865 *)
p[n_] := If[MemberQ[u, w[[n]]], 0, 1];
t = Table[p[n], {n, 1, z}];
Flatten[Position[t, 0]];  (* A298866 *)
Flatten[Position[t, 1]];  (* A298867 *)

cp's OEIS Frontend

A257409 Values of n such that there are exactly 2 solutions to x^2 - y^2 = n, with x > y >= 0.

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A257411 Values of n such that there are exactly 4 solutions to x^2 - y^2 = n with x > y >= 0.

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A257417 Values of n such that there are exactly 10 solutions to x^2 - y^2 = n with x > y >= 0.

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A257410 Values of n such that there are exactly 3 solutions to x^2 - y^2 = n with x > y >= 0.

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A257412 Values of n such that there are exactly 5 solutions to x^2 - y^2 = n with x > y >= 0.

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A257413 Values of n such that there are exactly 6 solutions to x^2 - y^2 = n with x > y >= 0.

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A257414 Values of n such that there are exactly 7 solutions to x^2 - y^2 = n with x > y >= 0.

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A257415 Values of n such that there are exactly 8 solutions to x^2 - y^2 = n with x > y >= 0.

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