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A132281 Noncomposites in A067200. Noncomposites (0, 1) and primes p such that A084380(p) = p^3 + 2 is prime.

%I A132281 #18 Jan 11 2020 05:54:15
%S A132281 0,1,3,5,29,71,83,113,173,263,311,419,431,491,503,509,683,701,761,773,
%T A132281 839,911,953,1031,1091,1103,1151,1193,1259,1283,1373,1451,1523,1583,
%U A132281 1601,1733,1823,1889,1931,2099,2153,2213,2273,2339,2351,2441,2531,2543
%N A132281 Noncomposites in A067200. Noncomposites (0, 1) and primes p such that A084380(p) = p^3 + 2 is prime.
%C A132281 The corresponding near-cube primes are A132282. Analog of near-square primes. After a(1) = 0, all values must be odd. Numbers of the form n^2+2 for n=1, 2, ... are 3, 6, 11, 18, 27, 38, 51, 66, 83, 102, ... (A059100). These are prime for indices n = 1, 3, 9, 15, 21, 33, 39, 45, 57, 81, 99, ... (A067201), corresponding to the near-square primes 3, 11, 83, 227, 443, 1091, 1523, 2027, ... (A056899). Helfgott proves with minor conditions that: "Let f be a cubic polynomial. Then there are infinitely many primes p such that f(p) is squarefree." Note that 47^3 + 2 = 103825 = 5^2 * 4153 and similarly 97^3 + 2 is divisible by 5^2, but otherwise an infinite number of p^3+2 are squarefree.
%H A132281 Amiram Eldar, <a href="/A132281/b132281.txt">Table of n, a(n) for n = 1..10000</a>
%H A132281 Harald Andres Helfgott, <a href="https://arxiv.org/abs/0706.1497">Power-free values, repulsion between points, differing beliefs and the existence of error</a>, arXiv:0706.1497 [math.NT], 2007.
%F A132281 {p in A000040 such that A067200(p) = A084380(p) = p^3 + 2 is in A000040}.
%F A132281 Union of {0,1} and A048637. - _R. J. Mathar_, Oct 18 2007
%e A132281 a(1) = 0 because 0^3 + 2 = 2 is prime and 0 is noncomposite.
%e A132281 a(2) = 1 because 1^3 + 2 = 5 is prime and 1 is noncomposite.
%e A132281 a(3) = 3 because 3^3 + 2 = 29 is prime and 3 is prime.
%e A132281 a(4) = 5 because 5^3 + 2 = 127 is prime and 5 is prime.
%e A132281 a(5) = 29 because 29^3 + 2 = 24391 is prime.
%e A132281 45 is not in the sequence because, although 45^3 + 2 = 91127 is prime, 45 is not prime.
%e A132281 63 is not in the sequence because, although 63^3 + 2 = 250049 is prime, 63 is not prime.
%e A132281 65 is not in the sequence because, although 65^3 + 2 = 274627 is prime, 65 is not prime.
%e A132281 a(6) = 71 because 71^3 + 2 = 357913 is prime.
%e A132281 a(7) = 83 because 83^3 + 2 = 571789 is prime.
%e A132281 a(8) = 113 because 113^3 + 2 = 1442899 is prime.
%e A132281 123 is not in the sequence because, although 123^3 + 2 = 1860869 is prime, 123 is not prime.
%Y A132281 Cf. A000040, A056899, A059100, A067200, A067201, A084380, A132282.
%K A132281 easy,nonn
%O A132281 1,3
%A A132281 _Jonathan Vos Post_, Aug 16 2007
%E A132281 More terms from _R. J. Mathar_, Oct 18 2007