A268518 Primes p == -1 mod 6 such that there are consecutive primes q and r with p^2 = q + r + 1.

5, 11, 17, 29, 53, 101, 113, 137, 179, 251, 281, 419, 431, 449, 521, 569, 599, 677, 797, 857, 941, 1049, 1091, 1163, 1181, 1259, 1289, 1451, 1721, 1901, 1907, 2087, 2213, 2339, 2351, 2447, 2531, 2549, 2729, 2801, 2957, 2963, 3137, 3251, 3323, 3593, 3659, 3761, 3821, 3863, 4049, 4133, 4217

Offset

1,1

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Marius Coman, Sequences of Integers, Conjectures and New Arithmetical Tools, Education Publishing, Columbus, Ohio, 2015. See page 7.

David Consiglio, Jr., Python Program

Example

5^2 = 11 + 13 + 1;
11^2 = 59 + 61 + 1;
17^2 = 139 + 149 + 1;
29^2 = 419 + 421 + 1;
53^2 = 1399 + 1409 + 1;
101^2 = 5099 + 5101 + 1;
113^2 = 6379 + 6389 + 1;
137^2 = 9377 + 9391 + 1;
179^2 = 16007 + 16033 + 1;
251^2 = 31489 + 31511 + 1;
281^2 = 39461 + 39499 + 1;
...

Maple

filter:= proc(n) local q, m;
if not isprime(n) then return false fi;
m:= (n^2-1)/2;
q:= prevprime(m);
nextprime(m-1)=2*m-q;
end proc:
select(filter, [seq(i, i=5..10000, 6)]); # Robert Israel, Aug 19 2020

Mathematica

p[n_]:=Sqrt[n+NextPrime[n]+1]; p[Select[Prime[Range[10^6]], PrimeQ[p[#]]&&Mod[p[#]+1, 6]==0&]] (* Ivan N. Ianakiev, Feb 10 2016 *)

Prog

(PARI) lista(nn) = {my(q = prime(1), r = prime(2)); for (n=1, nn, s = q + r + 1; if (issquare(s, &p) && isprime(p) && ((p % 6)==5), print1(p, ", ")); q = r; r = nextprime(r+1); ); } \\ Michel Marcus, Feb 11 2016

Crossrefs

Subsequence of A092740.

Sequence in context: A162336 A234346 A074267 * A268521 A230138 A231652

Adjacent sequences: A268515 A268516 A268517 * A268519 A268520 A268521

Keyword

nonn

Author

N. J. A. Sloane, Feb 10 2016

Extensions

Corrected and extended by David Consiglio, Jr., Feb 10 2016

a(28)-a(53) from Ivan N. Ianakiev, Feb 10 2016

Status

approved