1,1

The corresponding near-cube prime indices q are A132281. Analogue of near-square primes. After a(1) = 1, 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 square-free." Note that 47^3 + 2 = 103825 = 5^2 * 4153, and similarly 97^3 + 2 is divisble by 5^2, but otherwise an infinite number of p^3+2 are square-free.

Harald Andres Helfgott, Power-free values, repulsion between points, differing beliefs and the existence of error

a(n) = A132281(n)^3 + 2. {p in A000040 such that for some q = 0, 1, or q in A000040, we have p = A067200(q) = A084380(q) = q^3 + 2 is in A000040}.

a(1) = 0^3 + 2 = 2 is prime, and 0 is noncomposite.

a(2) = 1^3 + 2 = 5 is prime, and 1 is noncomposite.

a(3) = 3^3 + 2 = 29 is prime, and 3 is prime.

a(4) = 5^3 + 2 = 127 is prime, and 5 is prime.

a(5) = 29^3 + 2 = 24391 is prime, and 29 is prime.

45^3 + 2 = 91127 is prime, but not in this sequence because 45 is not prime.

63^3 + 2 = 250049 is prime, but not in this sequence because 63 is not prime.

a(6) = 71^3 + 2 = 357913 is prime.

a(7) = 83^3 + 2 = 571789 is prime.

a(8) = 113^3 + 2 = 1442899 is prime.

Join[{2, 5}, Select[Prime[Range[200]]^3 + 2, PrimeQ[ # ] &]] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Aug 17 2007

Cf. A000040, A056899, A059100, A067200, A067201, A084380, A132281.

Sequence in context: A057794 A073715 A104083 this_sequence A007014 A098682 A108367

Adjacent sequences: A132279 A132280 A132281 this_sequence A132283 A132284 A132285

easy,nonn

Jonathan Vos Post (jvospost3(AT)gmail.com), Aug 16 2007

More terms from Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), Aug 17 2007