cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A051859 Values of C (the hypotenuse) of a Pythagorean triangle with A (the short leg) and C both prime and part of a twin prime.

Original entry on oeis.org

5, 13, 61, 181, 421, 5101, 60901, 135721, 161881, 163021, 218461, 595141, 1108561, 2574181, 2740141, 3248701, 3535141, 3723721, 3729181, 8197201, 13933921, 20218441, 23605321, 28569241, 33874681, 47248921, 68667481, 69372421
Offset: 1

Views

Author

Stuart M. Ellerstein (ellerstein(AT)aol.com), Dec 14 1999

Keywords

Comments

All terms of the sequence must be part of a Pythagorean triple of the form (2u-1), 2u*(u-1), (2u^2 - 2u + 1). - Joshua Zucker, May 12 2006

Links

Crossrefs

See A051642 for the A's and A051858 for the B's.
Subset of A067756.

Programs

  • Mathematica
    tppQ[{a_,b_,c_}]:=AllTrue[{a,c},PrimeQ]&&AnyTrue[a+{2,-2},PrimeQ] && AnyTrue[ c+{2,-2},PrimeQ]; Select[Table[{2n-1,2n(n-1),2n^2-2n+1},{n,2,10000}],tppQ][[All,3]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 27 2021 *)

Formula

A051858 + 1, or 2*A051891^2 - 2*A051891 + 1, or 2*A051892^2 + 2*A051892 + 1. - Joshua Zucker, May 12 2006

Extensions

More terms from Joshua Zucker, May 12 2006

A051858 Values of B (the even leg) of a Pythagorean triangle with A and C both prime and part of a twin prime.

Original entry on oeis.org

4, 12, 60, 180, 420, 5100, 60900, 135720, 161880, 163020, 218460, 595140, 1108560, 2574180, 2740140, 3248700, 3535140, 3723720, 3729180, 8197200, 13933920, 20218440, 23605320, 28569240, 33874680, 47248920, 68667480, 69372420
Offset: 1

Views

Author

Stuart M. Ellerstein (ellerstein(AT)aol.com), Dec 14 1999

Keywords

Comments

All terms of the sequence must be part of a Pythagorean triple of the form (2u-1), 2u*(u-1), (2u^2 - 2u + 1). - Joshua Zucker, May 11 2006

Crossrefs

See A051642 for the A's and A051859 for the C's.

Extensions

a(11) corrected by the author, Jun 03 2002
More terms from Joshua Zucker, May 11 2006

A051892 Values of e, the lesser key or generating number for Pythagorean triangles in which S (the odd short leg) and U (the hypotenuse) are twin primes.

Original entry on oeis.org

1, 2, 5, 9, 14, 50, 174, 260, 284, 285, 330, 545, 744, 1134, 1170, 1274, 1329, 1364, 1365, 2024, 2639, 3179, 3435, 3779, 4115, 4860, 5859, 5889, 6035, 6669, 8645, 8960, 8979, 9714, 9944, 10115, 10179, 15770, 16589, 16875, 19325, 20214, 20265, 21450
Offset: 1

Views

Author

Stuart M. Ellerstein (ellerstein(AT)aol.com), Dec 17 1999

Keywords

Links

Crossrefs

Equals A051891 - 1. See A051642 for the S's, A051859 for the U's and A051858 for the T's (the even long leg).

Extensions

More terms from Joshua Zucker, May 12 2006

A284034 Primes p such that (p^2 - 3)/2 and (p^2 + 1)/2 are twin primes.

Original entry on oeis.org

3, 5, 11, 19, 29, 79, 101, 349, 409, 449, 521, 569, 571, 661, 739, 991, 1091, 1129, 1181, 1459, 1489, 1531, 1901, 2269, 2281, 2341, 2351, 2389, 2549, 2659, 2671, 2719, 2729, 2731, 3109, 4049, 4349, 5279, 5431, 5471, 5531, 5591, 5669, 6329, 6359, 6871, 7559, 7741
Offset: 1

Views

Author

Giuseppe Coppoletta, Mar 19 2017

Keywords

Comments

Primes which correspond to the short leg of an integral right triangle whose hypotenuse is part of a twin prime pair.
Each term p of the sequence must be part of a Pythagorean triple of the form {p, (p^2 - 1)/2, (p^2 + 1)/2} corresponding to {a(n), A284035(n) - 1, A284035(n)}.

Examples

			The prime p = 79 is in the sequence because (p^2-3)/2 = 3119 and (p^2+1)/2 = 3121 are twin primes. Remark that {79, 3120, 3121} is a Pythagorean triple.

Links

Crossrefs

Cf. A284035, A048161, A165635, A284036, A051642.

Programs

  • Mathematica
    Select[Prime@ Range[10^3], Function[p, Times @@ Boole@ Map[PrimeQ[(p^2 + #)/2 ] &, {-3, 1}] == 1]] (* Michael De Vlieger, Mar 20 2017 *)
Select[Prime[Range[1000]],AllTrue[{(#^2-3)/2,(#^2+1)/2},PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 04 2017 *)
  • PARI
    isok(p) = isprime(p) && isprime((p^2-3)/2) && isprime((p^2+1)/2); \\ Michel Marcus, Mar 31 2017
  • Sage
    [p for p in prime_range(10000) if is_prime((p^2-3)//2) and is_prime((p^2+1)//2)]

A051891 Values of m, the main key or generating number for Pythagorean triangles in which S (the odd short leg) and U (the hypotenuse) are twin primes.

Original entry on oeis.org

2, 3, 6, 10, 15, 51, 175, 261, 285, 286, 331, 546, 745, 1135, 1171, 1275, 1330, 1365, 1366, 2025, 2640, 3180, 3436, 3780, 4116, 4861, 5860, 5890, 6036, 6670, 8646, 8961, 8980, 9715, 9945, 10116, 10180, 15771, 16590, 16876, 19326, 20215, 20266, 21451
Offset: 1

Views

Author

Stuart M. Ellerstein (ellerstein(AT)aol.com), Dec 17 1999

Keywords

Links

Crossrefs

Cf. A051892. See A051642 for the S's, A051859 for the U's and A051858 for the T's (the even long leg).

Formula

a(n) = (A051642(n) + 1) / 2. - Sean A. Irvine, Oct 12 2021

Extensions

More terms from Joshua Zucker, May 12 2006
Showing 1-5 of 5 results.

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