A286679 - OEIS

%I #35 Sep 08 2022 08:46:19

%S 17,33,81,113,193,241,353,561,641,913,1121,1233,1473,1873,2321,2481,

%T 2993,3361,3553,4161,4593,5281,6273,6801,7073,7633,7921,8513,10753,

%U 11441,12513,12881,14801,15201,16433,17713,18593,19953,21361,21841

%N Numbers of the form (2*prime(n)^2 + 1)/3.

%C 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.

%H G. C. Greubel, <a href="/A286679/b286679.txt">Table of n, a(n) for n = 3..1000</a>

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

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

%F 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.

%t (2Prime[Range[3, 50]]^2 + 1)/3 (* _Alonso del Arte_, May 12 2017 *)

%o (PARI) {

%o forprime(n=5,300,

%o          print1((2*n^2+1)/3", ")

%o         )

%o }

%o (Magma) [(2*NthPrime(n)^2+1)/3: n in [3..50]]; // _Vincenzo Librandi_, May 15 2017 *)

%Y Cf. A075888, A001318.

%K nonn

%O 3,1

%A _Dimitris Valianatos_, May 12 2017