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

0, 1, 3, 5, 29, 71, 83, 113, 173, 263, 311, 419, 431, 491, 503, 509, 683, 701, 761, 773, 839, 911, 953, 1031, 1091, 1103, 1151, 1193, 1259, 1283, 1373, 1451, 1523, 1583, 1601, 1733, 1823, 1889, 1931, 2099, 2153, 2213, 2273, 2339, 2351, 2441, 2531, 2543

Offset: 1

Author

Jonathan Vos Post, Aug 16 2007

Keywords

• easy
• nonn

Comments

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.

Examples

			a(1) = 0 because 0^3 + 2 = 2 is prime and 0 is noncomposite.
a(2) = 1 because 1^3 + 2 = 5 is prime and 1 is noncomposite.
a(3) = 3 because 3^3 + 2 = 29 is prime and 3 is prime.
a(4) = 5 because 5^3 + 2 = 127 is prime and 5 is prime.
a(5) = 29 because 29^3 + 2 = 24391 is prime.
45 is not in the sequence because, although 45^3 + 2 = 91127 is prime, 45 is not prime.
63 is not in the sequence because, although 63^3 + 2 = 250049 is prime, 63 is not prime.
65 is not in the sequence because, although 65^3 + 2 = 274627 is prime, 65 is not prime.
a(6) = 71 because 71^3 + 2 = 357913 is prime.
a(7) = 83 because 83^3 + 2 = 571789 is prime.
a(8) = 113 because 113^3 + 2 = 1442899 is prime.
123 is not in the sequence because, although 123^3 + 2 = 1860869 is prime, 123 is not prime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Harald Andres Helfgott, Power-free values, repulsion between points, differing beliefs and the existence of error, arXiv:0706.1497 [math.NT], 2007.

Crossrefs

Cf. A000040, A056899, A059100, A067200, A067201, A084380, A132282.

Formula

{p in A000040 such that A067200(p) = A084380(p) = p^3 + 2 is in A000040}.

Union of {0,1} and A048637. - R. J. Mathar, Oct 18 2007

Extensions

More terms from R. J. Mathar, Oct 18 2007