A286679 a(n) = (2*prime(n)^2 + 1)/3.

17, 33, 81, 113, 193, 241, 353, 561, 641, 913, 1121, 1233, 1473, 1873, 2321, 2481, 2993, 3361, 3553, 4161, 4593, 5281, 6273, 6801, 7073, 7633, 7921, 8513, 10753, 11441, 12513, 12881, 14801, 15201, 16433, 17713, 18593, 19953, 21361, 21841

Offset: 3

Author

Dimitris Valianatos, May 12 2017

Keywords

• nonn
• easy

Comments

For primes p other than 3, p == 1 or 2 (mod 3) and p^2 == 1 (mod 3). Thus 2*p^2 + 1 is a multiple of 3.

Links

• G. C. Greubel, Table of n, a(n) for n = 3..1000

Crossrefs

Cf. A075888, A001318.

Programs

Magma

[(2*NthPrime(n)^2+1)/3: n in [3..50]]; // Vincenzo Librandi, May 15 2017

Mathematica

(2Prime[Range[3, 50]]^2 + 1)/3 (* Alonso del Arte, May 12 2017 *)

PARI

{
forprime(n=5,300,
         print1((2*n^2+1)/3", ")
        )
}

Formula

Product_{n >= 3} (3*a(n) + 1) / (3*a(n) - 1) = (26/25) * (50/49) * (122/121) * ... = 54/(5*Pi^2) = 1.0942687833372479315938982026650585002 (constant).

a(3) = 17; a(n + 1) = a(n) + 16 * A075888(n-2) for n > 3.

Numbers of the form 16k + 1 for some k. In particular, k belongs to A001318, excluding those for which sqrt(24 * A001318(k) + 1) are composites.