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.

A067756 Prime hypotenuses of Pythagorean triangles with a prime leg.

Original entry on oeis.org

5, 13, 61, 181, 421, 1741, 1861, 2521, 3121, 5101, 8581, 9661, 16381, 19801, 36721, 60901, 71821, 83641, 100801, 106261, 135721, 161881, 163021, 199081, 205441, 218461, 273061, 282001, 337021, 388081, 431521, 491041, 531481, 539761, 552301
Offset: 1

Views

Author

Henry Bottomley, Jan 31 2002

Keywords

Comments

Apart from the first two terms, every term is congruent to 1 modulo 60 and is of the form 450k^2 +- 30k + 1 or 450k^2 +- 330k + 61 for some k.
Every term of the sequence after the second is a prime p congruent to 1 (mod 60), i.e., for n > 2, a(n) is a subsequence of A088955. The Pythagorean triple is {sqrt(2p-1), p-1, p}. - Lekraj Beedassy, Mar 12 2002
Primes p such that 2*p-1 is the square of a prime. - Robert Israel, Sep 16 2014
Primes p of the form ((q+1)/2)^2 + ((q-1)/2)^2, where q is a prime; then q belongs to A048161. - Thomas Ordowski, May 22 2015
The other (i.e., long) leg of the Pythagorean triangle is p-1. - Zak Seidov, Oct 30 2015

Examples

			For a(1)=5, the right triangle is 3, 4, 5 with 3 and 5 prime.
For a(10)=5101, the right triangle is 101, 5100, 5101 with 101 and 5101 prime.

Links

Crossrefs

Contains every value of A051859.

Programs

  • Maple
    N:= 10^8: # to get all terms <= N
Primes:= select(isprime,[$3..floor(sqrt(2*N-1))]):
f:= proc(p) local q; q:= (p^2+1)/2; if isprime(q) then q else NULL fi end proc:
map(f, Primes); # Robert Israel, Sep 16 2014
  • Mathematica
    f[n_]:=((p-1)/2)^2+((p+1)/2)^2; lst={};Do[p=Prime[n];If[PrimeQ[f[p]],AppendTo[lst,f[p]]],{n,6!}];lst (* Vladimir Joseph Stephan Orlovsky, Jul 27 2009 *)
  • PARI
    forprime(p=3,10^3,if(isprime(q=(p^2+1)/2),print1(q,", "))) \\ Derek Orr, Apr 30 2015

Formula

a(n) = (A048161(n)^2 + 1)/2 = A067755(n) + 1.

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

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

A284035 Upper twin primes which correspond to the hypotenuse in a Pythagorean triple whose short leg is also a prime.

Original entry on oeis.org

5, 13, 61, 181, 421, 3121, 5101, 60901, 83641, 100801, 135721, 161881, 163021, 218461, 273061, 491041, 595141, 637321, 697381, 1064341, 1108561, 1171981, 1806901, 2574181, 2601481, 2740141, 2763601, 2853661, 3248701, 3535141, 3567121, 3696481, 3723721, 3729181, 4832941
Offset: 1

Views

Author

Giuseppe Coppoletta, Mar 19 2017

Keywords

Comments

A284034 gives the corresponding short leg primes in the definition.

Examples

			The prime q = 3121 is in the sequence because q - 2 = 3119 is prime and {79, 3120, 3121} is a Pythagorean triple with prime short leg (see example in A284034).

Crossrefs

Cf. A051859, A067756, A284034.

Programs

  • PARI
    lista(nn) = forprime(p=3, nn, if (isprime(p) && isprime((p^2-3)/2) && isprime(q=(p^2+1)/2), print1(q, ", "))); \\ Michel Marcus, Apr 01 2017

Formula

A284034(n)^2 + (a(n) - 1)^2 = a(n)^2, i.e., a(n) = (A284034(n)^2 + 1)/2.

Extensions

More terms from Michel Marcus, Apr 01 2017
Showing 1-5 of 5 results.

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