|
%I #75 Jul 26 2022 21:59:59
%S 2,3,5,17,41,59,71,89,101,131,167,173,293,383,677,701,743,761,773,827,
%T 839,857,911,1091,1097,1163,1181,1193,1217,1373,1427,1487,1559,1583,
%U 1709,1811,1847,1931,1973,2129,2273,2309,2339,2411,2663,2729,2789,2957
%N Primes p such that p^2 + p + 1 is prime.
%C 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
%C Starting with a(3)=5 all terms are of the form 6k-1, k in A147683. - _Zak Seidov_, Nov 10 2008
%C 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
%C The generated prime numbers p^2 + p + 1 are exactly A053183. - _Bernard Schott_, Dec 20 2012
%C Positive squarefree k such that the sum of divisors of k^2 is prime. - _Peter Munn_, Feb 02 2018
%H Jon E. Schoenfield, <a href="/A053182/b053182.txt">Table of n, a(n) for n = 1..10000</a> (first 2650 terms from M. F. Hasler).
%H Paul T. Bateman and Roger A. Horn, <a href="http://dx.doi.org/10.1090/S0025-5718-1962-0148632-7">A heuristic asymptotic formula concerning the distribution of prime numbers</a>, Math. Comp. 16 (1962), 363-367.
%H Paolo Santonastaso and Ferdinando Zullo, <a href="https://arxiv.org/abs/2012.14861">Linearized trinomials with maximum kernel</a>, arXiv:2012.14861 [math.NT], 2020.
%t Select[Prime[Range[427]], PrimeQ[#^2+#+1]&] (* _Bruno Berselli_, Nov 08 2011 *)
%o (PARI) isA053182(n)=isprime(n) && isprime(n^2+n+1) \\ _Michael B. Porter_, Apr 23 2010
%o (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
%o (Magma) [p: p in PrimesUpTo(10000) | IsPrime(p^2+p+1)]; // _Vincenzo Librandi_, Aug 06 2010
%Y Cf. A053184, A065508, A091567, A147683, A188596.
%K easy,nonn
%O 1,1
%A _Enoch Haga_, Mar 01 2000
%E List changed to cross-reference by _Franklin T. Adams-Watters_, May 12 2010
|
