A088955 -id:A088955 - cp's OEIS frontend

A067756 Prime hypotenuses of Pythagorean triangles with a prime leg.

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

Henry Bottomley, Jan 31 2002

nonn

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

			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.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 184 terms from Andreas Boe)
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Contains every value of A051859.

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

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 *)

forprime(p=3,10^3,if(isprime(q=(p^2+1)/2),print1(q,", "))) \\ Derek Orr, Apr 30 2015

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

A088958 Numbers n such that 60*n+1 is prime.

Original entry on oeis.org

1, 3, 4, 7, 9, 10, 11, 17, 20, 22, 23, 27, 29, 30, 31, 36, 37, 38, 39, 42, 50, 51, 52, 53, 55, 56, 59, 67, 70, 71, 74, 76, 77, 80, 81, 85, 88, 92, 93, 94, 95, 97, 98, 102, 105, 106, 107, 108, 111, 113, 114, 116, 122, 126, 127, 128, 129, 135, 136, 137, 141, 142, 143, 144

Offset: 1

Lekraj Beedassy, Dec 01 2003

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5700

Cf. A088955, A090605.

[n: n in [0..200] | IsPrime(60*n+1)]; // Vincenzo Librandi, May 23 2017

Select[Range[200], PrimeQ[60 # + 1] &] (* Vincenzo Librandi, May 23 2017 *)

isok(n) = isprime(60*n+1) \\ Michel Marcus, Jul 27 2013

a(n) = (A088955(n)-1)/60 = (A000040(A090605(n))-1)/60.

More terms from Ray Chandler, Dec 02 2003

A090605 Numbers m such that m-th prime is of the form 60*k+1.

Original entry on oeis.org

18, 42, 53, 82, 100, 110, 121, 172, 197, 216, 221, 257, 271, 279, 284, 326, 331, 339, 347, 369, 431, 438, 445, 450, 464, 474, 496, 556, 575, 585, 603, 618, 624, 647, 651, 682, 701, 730, 737, 741, 751, 764, 775, 798, 820, 829, 835, 841, 859, 873, 881, 894

Offset: 1

Ray Chandler, Dec 06 2003

nonn

A088955 indexed by A000040.

The asymptotic density of this sequence is 1/16 (by Dirichlet's theorem). - Amiram Eldar, Mar 01 2021

Cf. A000040, A088955, A088958.

Select[Range[900], Mod[Prime[#], 60] == 1 &] (* Amiram Eldar, Mar 01 2021 *)

a(n) = k such that A000040(k) = A088955(n) = 60*A088958(n)+1.

A263770 Smallest prime q such that (prime(n)^2 + q*prime(n))/(prime(n) + 1) is an integer.

Original entry on oeis.org

7, 5, 7, 17, 13, 29, 19, 41, 73, 31, 97, 191, 43, 89, 97, 109, 61, 311, 137, 73, 149, 241, 337, 181, 197, 103, 313, 109, 331, 229, 257, 397, 139, 281, 151, 457, 317, 821, 337, 349, 181, 547, 193, 389, 199, 401, 1061, 449, 229, 461, 937, 241, 727, 757, 1033, 1321, 271, 1361, 557

Offset: 1

Juri-Stepan Gerasimov, Oct 25 2015

nonn

Least prime q such that q == 1 (mod prime(n) + 1).

Consists of the initial terms of sequences A002476, A002144, A002476, A007519, A068228, A140444, A061237, A141881, A107008, A132230, A133870, A141868, A124826, A142292, A142398, A141948, A088955, A142003, ...

Cf. A263729, A263730, A263769.

Table[q = 2; While[! IntegerQ[(Prime[n]^2 + q Prime@ n)/(Prime@ n + 1)], q = NextPrime@ q]; q, {n, 59}] (* Michael De Vlieger, Oct 26 2015 *)

a(n) = {p = prime(n); q = 2; while ((p^2 + p*q) % (p + 1), q = nextprime(q+1)); q;} \\ Michel Marcus, Oct 26 2015

5 is in this sequence because (prime(2)^2 + 5*prime(2))/(prime(2) + 1) = 6 and 5 is prime.

A142786 Primes congruent to 7 mod 60.

Original entry on oeis.org

7, 67, 127, 307, 367, 487, 547, 607, 727, 787, 907, 967, 1087, 1327, 1447, 1567, 1627, 1747, 1867, 1987, 2287, 2347, 2467, 2647, 2707, 2767, 2887, 3067, 3187, 3307, 3547, 3607, 3727, 3847, 3907, 3967, 4027, 4327, 4447, 4507, 4567, 4987, 5107, 5167, 5227

Offset: 1

N. J. A. Sloane, Jul 11 2008

Comment from Joshua S.M. Weiner, Oct 12 2012 (Start)
Intersection of A068229 and A141882. Subsequence of A132231.
Congruence classes of primes mod 60: A088955 (1), (this sequence 7),  A117047 (11), A142787 (13), A142788 (17), A142789 (19), A142790 (23), A142791 (29), A142792 (31), A142793 (37), A142794 (41), A142795 (43), A142796 (47), A142797 (49), A142798 (53), A142799 (59). (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A142322.

[p: p in PrimesUpTo(6000) | p mod 60 eq 7 ]; // Vincenzo Librandi, Sep 04 2012

Select[Prime[Range[1000]], Mod[#, 60] == 7 &] (* T. D. Noe, Oct 12 2012 *)
Select[Range[7,5300,60],PrimeQ] (* Harvey P. Dale, Nov 21 2018 *)

A217692 Primes p such that p = 1 + 27720*k for some k.

Original entry on oeis.org

55441, 110881, 332641, 388081, 415801, 471241, 498961, 526681, 748441, 859321, 970201, 1025641, 1053361, 1108801, 1247401, 1275121, 1302841, 1358281, 1469161, 1580041, 1912681, 1940401, 1995841, 2051281, 2189881, 2273041, 2300761, 2383921, 2411641, 2855161

Offset: 1

Joshua S.M. Weiner, Oct 11 2012

This is a congruence class of a prime wheel factorization mod 27720. Note that 27720 is the LCM of {1,...,11}.

Bruno Berselli, Table of n, a(n) for n = 1..1000

Cf. A002476, A068228, A088955, A217587, A217588.

[p: p in PrimesUpTo(3*10^6) | IsOne(p mod 27720)]; // Bruno Berselli, Oct 12 2012

Select[Table[1 + 27720*k, {k, 200}], PrimeQ] (* T. D. Noe, Oct 11 2012 *)

cp's OEIS Frontend

A067756 Prime hypotenuses of Pythagorean triangles with a prime leg.

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A088958 Numbers n such that 60*n+1 is prime.

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A090605 Numbers m such that m-th prime is of the form 60*k+1.

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A263770 Smallest prime q such that (prime(n)^2 + q*prime(n))/(prime(n) + 1) is an integer.

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Mathematica

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A142786 Primes congruent to 7 mod 60.

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Magma

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A217692 Primes p such that p = 1 + 27720*k for some k.

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Magma

Mathematica