A116945 -id:A116945 - cp's OEIS frontend

A002731 Numbers k such that (k^2 + 1)/2 is prime.

3, 5, 9, 11, 15, 19, 25, 29, 35, 39, 45, 49, 51, 59, 61, 65, 69, 71, 79, 85, 95, 101, 121, 131, 139, 141, 145, 159, 165, 169, 171, 175, 181, 195, 199, 201, 205, 209, 219, 221, 231, 245, 261, 271, 275, 279, 289, 299, 309, 315, 321, 325, 329, 335, 345, 349, 371, 375, 379, 391, 399, 405

Offset: 1

N. J. A. Sloane

From Wolfdieter Lang, Feb 24 2012: (Start)
a(n) = sqrt(8*A129307(n)+1) = sqrt(2*A027862(n)-1), n >= 1.
a(n) is the nontrivial solution of the congruence a(n)^2 == 1 (Modd A027862(n)). The trivial one is +1. For Modd n see a comment on A203571. E.g., a(3)^2 = 81 == 1 (Modd 41), see a comment on A027862.
(End)

L. Euler, De numeris primis valde magnis (E283), reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 3, p. 24.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
L. Euler, De numeris primis valde magnis (E283), The Euler Archive.

Cf. A027861. A027862 gives primes, A091277 gives prime indices.

Cf. A000982, A010051, A116945, A129307, A188546, A188547, A187431.

a002731 n = a002731_list !! (n-1)
a002731_list = filter ((== 1) . a010051 . a000982) [1, 3 ..]
-- Reinhard Zumkeller, Jul 13 2014

[n: n in [3..410] | IsPrime((n^2+1) div 2) ]; // Vincenzo Librandi, Sep 25 2012

Select[Range[400], PrimeQ[(#^2 + 1)/2] &] (* Alonso del Arte, Feb 24 2012 *)

forstep(n=1,10^3,2, if(isprime((n^2+1)/2),print1(n,", ")));
/* Joerg Arndt, Sep 02 2012 */

a(n) = 2*A027861(n) + 1.

A048270 Sequence of 2 Pythagorean triangles, each with a leg and hypotenuse prime. The leg of the second triangle is the hypotenuse of the first.

Original entry on oeis.org

3, 11, 19, 59, 271, 349, 521, 929, 1031, 1051, 1171, 2381, 2671, 2711, 2719, 3001, 3499, 3691, 4349, 4691, 4801, 4999, 5591, 5669, 6101, 6359, 6361, 7159, 7211, 7489, 8231, 8431, 8761, 9241, 10099, 10139, 11719, 11821, 12239, 12281, 12781

Offset: 1

Harvey Dubner (harvey(AT)dubner.com)

nonn

It is conjectured that there are infinitely many such pairs of triangles.

Subsequence of A048161. - Lekraj Beedassy, Sep 16 2005

			p(1)=3 because 3 is prime, 5 = (3*3 + 1)/2 and 13 = (5*5 + 1)/2, 5, 13 both prime.

Ray Chandler, Table of n, a(n) for n = 1..10000
H. Dubner and T. Forbes, Prime Pythagorean triangles, J. Integer Seqs., Vol. 4 (2001), #01.2.3.

Cf. A048161, A048295, A308635, A308636. Primes in A116945.

For each p(n), there is a q=(p*p+1)/2 and r=(q*q+1)/2 such that p, q, r are all prime.

More terms from Ray Chandler, Jun 12 2019

A187431 Numbers n that generate 5 primes under the first 5 iterations of the map n->A002731(n).

Original entry on oeis.org

30131199, 50817201, 56496039, 74316929, 171407609, 276672151, 293315671, 337876949, 356498179, 359830101, 372590921, 432448789, 501182201, 541961069, 577016839, 616411051, 749536461, 776113741, 903321909, 919203811, 1005047121, 1285328811, 1323139751, 1340738371

Offset: 1

Zak Seidov, Apr 05 2011

nonn

Numbers n such that m=(n^2+1)/2, p=(m^2+1)/2, q=(p^2+1)/2, r=(q^2+1)/2 and s=(r^2+1)/2 are all prime.
Subsequence of A188547 which itself is subsequence of A188546 which is subsequence of A116945.
a(1)=30131199=A188547(70).
Two numbers n that generate 6 primes... are a(23)=1323139751 and a(78)=10185588801.

Charles R Greathouse IV, Table of n, a(n) for n = 1..405, replacing an earlier b-file from Zak Seidov
Stephan Baier, On the Bateman-Horn conjecture, Journal of Number Theory 96, 432-448 (2002).
Paul T. Bateman, Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962) 363-367.

Cf. A002731, A105318, A116945, A188546, A188547.

A188546 Numbers n such that m=(n^2+1)/2, p=(m^2+1)/2 and q=(p^2+1)/2 are all prime.

Original entry on oeis.org

69, 271, 349, 3001, 3399, 4949, 6051, 9101, 9751, 10099, 10149, 11719, 12281, 15911, 22569, 24269, 25201, 25889, 28841, 31979, 37271, 39901, 42109, 44929, 46399, 48321, 50811, 60009, 63659, 63999, 71051, 71851, 75089, 76711, 87029, 96791, 103701, 110551, 111411, 112461, 113949, 125721, 126089, 127959, 129261, 131859, 132939, 137481, 144651

Offset: 1

Zak Seidov, Apr 03 2011

nonn

a(1) = 69 = A116945(5).
Numbers n that generate three primes under the first three iterations of the map n-> A002731(n).
Subsequence of A116945.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002731, A105318, A116945, A188547, A187431.

r:=func< k | (k^2+1) div 2 >; [ n: n in [1..145000 by 2] | IsPrime(r(n)) and IsPrime(r(r(n))) and IsPrime(r(r(r(n)))) ];  // Bruno Berselli, Apr 05 2011

s={}; Do[If[PrimeQ[m=(n^2+1)/2] && PrimeQ[p=(m^2+1)/2] && PrimeQ[q=(p^2+1)/2], Print[n]; AppendTo[s,n]], {n,1,300000,2}]; s
mpqQ[n_]:=Module[{m=(n^2+1)/2,p},p=(m^2+1)/2;AllTrue[{m,p,(p^2+1)/2},PrimeQ]]; Select[Range[144700],mpqQ] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 18 2021 *)

v=vector(10^4);i=0;forstep(n=1,9e9,2,if(isprime(m=(n^2+1)/2)&isprime(p=(m^2+1)/2)&isprime(q=(p^2+1)/2),v[i++]=n;if(i==#v,return(v)))) \\ Charles R Greathouse IV, Apr 05 2011

A188547 Numbers n such that m=(n^2+1)/2, p=(m^2+1)/2, q=(p^2+1)/2, and r=(q^2+1)/2 are all prime.

Original entry on oeis.org

4949, 6051, 169219, 183241, 560769, 1113621, 1306689, 1370269, 1421869, 1485561, 1640711, 1730709, 1876351, 1967769, 2147661, 2217351, 2293939, 2428461, 2440871, 3346661, 3625139, 3625889, 3763969, 3991209, 4020711, 4728141, 5219691, 5547221, 5554939, 5965699, 7345719, 8495879

Offset: 1

Zak Seidov, Apr 03 2011

nonn

a(1) = 4949 = A188546(6) = A116945(53).
Subsequence of A188546.
Numbers n which generate 4 primes under the first four iterations of the map n-> A002731(n).
Among first 10000 terms, there are 1072 primes, the first a(3) = 169219 and the last a(10000) = 16541600731. - Zak Seidov, Jan 16 2019

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000, replacing an earlier file from Zak Seidov

Cf. A002731, A105318, A116945, A188546, A187431.

r:=func< k | (k^2+1) div 2 >; [ n: n in [1..1000000 by 2] | IsPrime(r(n)) and IsPrime(r(r(n))) and IsPrime(r(r(r(n))))and IsPrime(r(r(r(r(n)))))]; // Vincenzo Librandi, Jan 16 2019

s={}; Do[If[PrimeQ[m=(n^2+1)/2] && PrimeQ[p=(m^2+1)/2] && PrimeQ[q=(p^2+1)/2] && PrimeQ[r=(q^2+1)/2], AppendTo[s,n]], {n,1,10000000,2}]; s

v=vector(10^4); i=0; forstep(n=1, 9e99, 2, if(isprime(m=(n^2+1)/2) && isprime(p=(m^2+1)/2) && isprime(q=(p^2+1)/2) && isprime(r=(q^2+1)/2), v[i++]=n; if(i==#v, return))) \\ Charles R Greathouse IV, Apr 12 2011

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A002731 Numbers k such that (k^2 + 1)/2 is prime.

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A048270 Sequence of 2 Pythagorean triangles, each with a leg and hypotenuse prime. The leg of the second triangle is the hypotenuse of the first.

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A187431 Numbers n that generate 5 primes under the first 5 iterations of the map n->A002731(n).

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A188546 Numbers n such that m=(n^2+1)/2, p=(m^2+1)/2 and q=(p^2+1)/2 are all prime.

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A188547 Numbers n such that m=(n^2+1)/2, p=(m^2+1)/2, q=(p^2+1)/2, and r=(q^2+1)/2 are all prime.

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