A053182 Primes p such that p^2 + p + 1 is prime.

2, 3, 5, 17, 41, 59, 71, 89, 101, 131, 167, 173, 293, 383, 677, 701, 743, 761, 773, 827, 839, 857, 911, 1091, 1097, 1163, 1181, 1193, 1217, 1373, 1427, 1487, 1559, 1583, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663, 2729, 2789, 2957

Offset

1,1

Comments

Roger Horn computed the first 776 terms of this sequence around 1961 to test (with Paul Bateman) their conjecture on the density of simultaneous primes in polynomials. - Charles R Greathouse IV, Apr 05 2011

Starting with a(3)=5 all terms are of the form 6k-1, k in A147683. - Zak Seidov, Nov 10 2008

Primes p such that the sum of divisors of p^2 (sigma(p^2) = A000203(p^2) = p^2+p+1) is prime. - Claudio Meller, Apr 07 2011

The generated prime numbers p^2 + p + 1 are exactly A053183. - Bernard Schott, Dec 20 2012

Positive squarefree k such that the sum of divisors of k^2 is prime. - Peter Munn, Feb 02 2018

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 2650 terms from M. F. Hasler).

Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363-367.

Paolo Santonastaso and Ferdinando Zullo, Linearized trinomials with maximum kernel, arXiv:2012.14861 [math.NT], 2020.

Mathematica

Select[Prime[Range[427]], PrimeQ[#^2+#+1]&] (* Bruno Berselli, Nov 08 2011 *)

Prog

(PARI) isA053182(n)=isprime(n) && isprime(n^2+n+1)  \\ Michael B. Porter, Apr 23 2010
(PARI) c=0; forprime(p=1, default(primelimit), isprime(p^2+p+1) & write("/tmp/b053182.txt", c++, " "p))  \\ M. F. Hasler, Apr 07 2011
(Magma) [p: p in PrimesUpTo(10000) | IsPrime(p^2+p+1)]; // Vincenzo Librandi, Aug 06 2010

Crossrefs

Cf. A053184, A065508, A091567, A147683, A188596.

Sequence in context: A189536 A163588 A270539 * A348153 A211972 A339855

Adjacent sequences: A053179 A053180 A053181 * A053183 A053184 A053185

Keyword

easy,nonn

Author

Enoch Haga, Mar 01 2000

Extensions

List changed to cross-reference by Franklin T. Adams-Watters, May 12 2010

Status

approved