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%I #39 Feb 21 2022 20:31:27
%S 1,2,3,5,6,8,9,12,15,17,18,21,29,30,32,39,41,42,44,48,53,54,56,60,69,
%T 71,74,77,83,87,95,102,104,108,116,117,120,126,131,135,143,144,146,
%U 152,153,155,162,168,177,179,180,186,191,207,212,219,221,225,239,240,243
%N Numbers m such that m^3 + m + 1 is prime.
%C For s = 5, 8, 11, 14, 17, 20, ... (A016789(s) for s>=2), m_s = 1 + m + m^s is composite for m>1. Also for m=1, m_s = 3 is a prime for any s. Here we consider the case s=3.
%C If m == 1 (mod 3), m_s == 0 (mod 3) for any s and is not prime for m > 1. Thus for n > 1, a(n) !== 1 (mod 3) and this is true for any similar sequence based on another s value (A002384, A049408, A075723). - _Jean-Christophe Hervé_, Sep 20 2014
%C Corresponding primes are in A095692.
%H T. D. Noe, <a href="/A049407/b049407.txt">Table of n, a(n) for n = 1..1000</a>
%e 3 is a term because 1 + 3 + 3^3 = 31 is a prime.
%p A049407:=n->`if`(isprime(n^3+n+1), n, NULL): seq(A049407(n), n=1..300); # _Wesley Ivan Hurt_, Nov 14 2014
%t Select[Range[500], PrimeQ[Total[#^Range[1, 3, 2]] + 1] &] (* _Vincenzo Librandi_, Jun 27 2014 *)
%o (PARI) is(n)=isprime(n^3+n+1) \\ _Charles R Greathouse IV_, Nov 20 2012
%o (Magma) [n: n in [0..300] | IsPrime(s) where s is 1+&+[n^i: i in [1..3 by 2]]]; // _Vincenzo Librandi_, Jun 27 2014
%o (Python)
%o from sympy import isprime
%o def ok(m): return isprime(m**3 + m + 1)
%o print([m for m in range(244) if ok(m)]) # _Michael S. Branicky_, Feb 17 2022
%Y Cf. A002384 (s=2), A049408 (s=4), A075723 (s=6).
%Y Cf. A095692 (corresponding primes).
%K nonn,easy,nice
%O 1,2
%A _N. J. A. Sloane_
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