A257412 -id:A257412 - cp's OEIS frontend

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

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

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

A257315 a(1) = 34; a(2) = 35; for n > 2, a(n) is the smallest number of the form prime + 32 not already used which shares a factor with a(n-1).

Original entry on oeis.org

34, 35, 45, 39, 51, 63, 49, 91, 105, 55, 75, 69, 93, 99, 111, 37, 259, 133, 171, 129, 43, 301, 189, 135, 85, 115, 145, 195, 141, 159, 183, 61, 793, 169, 273, 213, 225, 205, 255, 231, 121, 363, 243, 261, 303, 309, 103, 1339, 325, 265, 295, 315, 339, 345, 369

Offset: 1

Vladimir Shevelev, Apr 20 2015

nonn

Analog of EKG-sequence (A064413) on the numbers of the form prime + 32.

Conjecture: the sequence {a(n)-32} is a permutation of primes (A000040).

Peter J. C. Moses, Table of n, a(n) for n = 1..1000

Cf. A000040, A064413, A257411, A257412, A257413, A257414.

f[n_] := Block[{o = 2^5, s, p, k}, s = {o + 2, o + 3}; For[k = 3, k <= n, k++, p = 2; While[GCD[p + o, s[[k - 1]]] == 1 || MemberQ[s, p + o], p = NextPrime@ p]; AppendTo[s, p + o]]; s]; f@ 55 (* Michael De Vlieger, Apr 20 2015 *)

More terms from Peter J. C. Moses, Apr 20 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

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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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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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A257315 a(1) = 34; a(2) = 35; for n > 2, a(n) is the smallest number of the form prime + 32 not already used which shares a factor with a(n-1).

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

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