A154777 -id:A154777 - cp's OEIS frontend

A155710 Intersection of A092572 and A154778: N = a^2 + 3b^2 = c^2 + 5d^2 for some positive integers a,b,c,d.

21, 36, 49, 61, 84, 109, 129, 144, 181, 189, 196, 201, 229, 241, 244, 301, 309, 324, 336, 349, 381, 409, 421, 436, 441, 469, 489, 516, 525, 541, 549, 576, 601, 661, 669, 709, 721, 724, 756, 769, 784, 804, 829, 849, 889, 900, 916, 921, 964, 976, 981, 1009, 1021

Offset: 1

M. F. Hasler, Jan 25 2009

Subsequence of A155570 (where a,b,c,d may be zero).

Cf. A000404, A154777, A092572, A097268, A154778, A155716, ...

isA155710(n,/* use optional 2nd arg to get other analogous sequences */c=[5,3]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,1111, isA155710(n) & print1(n","))

A201688 Primes of the form p^2 + 18, where p is prime.

Original entry on oeis.org

43, 67, 139, 307, 379, 547, 859, 1699, 1867, 3499, 3739, 4507, 5059, 5347, 6907, 10627, 11467, 18787, 29947, 32059, 32779, 39619, 49747, 57139, 58099, 66067, 72379, 73459, 78979, 80107, 96739, 97987, 109579, 120427, 134707, 151339, 157627, 187507, 218107

Offset: 1

Zak Seidov, Dec 03 2011

nonn

All terms = 3 mod 8.

Corresponding p's: 5, 7, 11, 17, 19, 23, 29, 41, 43, 59, 61, 67, 71, 73, 83, 103, 107, 137, 173, 179, 181, 199, 223, 239, 241, 257, 269, 271, 281, 283, 311, 313, 331, 347, 367, 389, 397, 433, 467, 479, 509. From these, 43, 67, 1699, 3499, 4507, 5059 are themselves of form prime^2+18.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

Cf. A154777, A201613.

Select[Table[Prime[x]^2+18, {x,100}], PrimeQ]

A201719 Primes of the form x^2 + 2y^2 such that y^2 + 2x^2 is also prime.

Original entry on oeis.org

11, 19, 43, 59, 67, 83, 107, 139, 163, 179, 211, 251, 307, 331, 419, 443, 467, 491, 563, 571, 587, 619, 643, 811, 883, 907, 947, 971, 1019, 1091, 1123, 1171, 1259, 1291, 1307, 1427, 1531, 1571, 1579, 1667, 1699, 1747, 1787, 1811, 1907, 1979, 1987, 2003, 2011

Offset: 1

Zak Seidov, Dec 04 2011

nonn

All terms == 3 mod 8 (cf. A007520).

			Corresponding pairs of primes:
(a(1),a(2))=(11,19): 11=3^2+2*1^2, 19=1^2+2*3^2
(a(3),a(4))=(43,59): 43=5^2+2*3^2, 59=3^2+2*5^2
(a(5),a(7))=(67,107): 67=7^2+2*3^2, 107=3^2+2*7^2.

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A154777.

With[{nn=50},Take[Union[Flatten[Select[{#[[1]]^2+2#[[2]]^2,2#[[1]]^2+ #[[2]]^2}&/@Subsets[Range[nn],{2}],And@@PrimeQ[#]&]]],nn]] (* Harvey P. Dale, Sep 15 2013 *)

A155571 Intersection of A000404, A092572 and A154778: N = a^2 + b^2 = c^2 + 3d^2 = e^2 + 5f^2 for some positive integers a,b,c,d,e,f.

Original entry on oeis.org

61, 109, 181, 229, 241, 244, 349, 409, 421, 436, 541, 549, 601, 661, 709, 724, 769, 829, 900, 916, 964, 976, 981, 1009, 1021, 1069, 1129, 1201, 1225, 1249, 1321, 1381, 1396, 1429, 1489, 1521, 1525, 1549, 1609, 1621, 1629, 1636, 1669, 1684, 1741, 1744, 1789

Offset: 1

M. F. Hasler, Jan 25 2009

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717, A155560-A155578.

isA155571(n,/* optional 2nd arg allows us to get other sequences */c=[5,3,1]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,1999, isA155571(n) & print1(n","))

A155713 Intersection of A154778 and A155716: N = a^2 + 5b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

Original entry on oeis.org

49, 70, 105, 145, 150, 166, 196, 214, 225, 241, 249, 280, 294, 321, 406, 409, 420, 441, 454, 505, 580, 600, 601, 609, 630, 664, 681, 694, 721, 726, 745, 769, 784, 841, 856, 870, 886, 889, 900, 934, 945, 964, 996, 1009, 1030, 1041, 1089, 1120, 1126, 1129

Offset: 1

M. F. Hasler, Jan 25 2009

Cf. A000404, A154777, A092572, A097268, A154778, A155716, A155717, A155560-A155578.

isA155713(n,/* optional 2nd arg allows us to get other sequences */c=[6,5]) = { for(i=1,#c, for(b=1,sqrtint((n-1)\c[i]), issquare(n-c[i]*b^2) & next(2)); return);1}
for( n=1,999, isA155713(n) & print1(n","))

A201544 Odd numbers of the form x^2 + 2*y^2 with positive integers x and y.

Original entry on oeis.org

3, 9, 11, 17, 19, 27, 33, 41, 43, 51, 57, 59, 67, 73, 75, 81, 83, 89, 97, 99, 107, 113, 121, 123, 129, 131, 137, 139, 147, 153, 163, 171, 177, 179, 187, 193, 201, 209, 211, 219, 225, 227, 233, 241, 243, 249, 251, 257, 267, 275, 281, 283, 289, 291, 297, 307

Offset: 1

Zak Seidov, Dec 02 2011

nonn

All terms == {1,3} mod 8. Terms that are not multiple of some previous term are prime numbers (see A033203, except for the first term 2 there).

For the numbers with positive proper representations see A225771 without member 1, the subsequence without 75 = 3*5^2, 147 = 3*7^2, 225 = (3*5)^2, 275 = 5^2*11, ... - Wolfdieter Lang, Jan 14 2025

Intersection of A005408 and A154777.

Cf. A033200 (primes), A033203, A225771.

cp's OEIS Frontend

A155710 Intersection of A092572 and A154778: N = a^2 + 3b^2 = c^2 + 5d^2 for some positive integers a,b,c,d.

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A201688 Primes of the form p^2 + 18, where p is prime.

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Mathematica

A201719 Primes of the form x^2 + 2y^2 such that y^2 + 2x^2 is also prime.

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Mathematica

A155571 Intersection of A000404, A092572 and A154778: N = a^2 + b^2 = c^2 + 3d^2 = e^2 + 5f^2 for some positive integers a,b,c,d,e,f.

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PARI

A155713 Intersection of A154778 and A155716: N = a^2 + 5b^2 = c^2 + 6d^2 for some positive integers a,b,c,d.

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PARI

A201544 Odd numbers of the form x^2 + 2*y^2 with positive integers x and y.

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