This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A286679 #38 Nov 23 2024 03:34:07
%S A286679 17,33,81,113,193,241,353,561,641,913,1121,1233,1473,1873,2321,2481,
%T A286679 2993,3361,3553,4161,4593,5281,6273,6801,7073,7633,7921,8513,10753,
%U A286679 11441,12513,12881,14801,15201,16433,17713,18593,19953,21361,21841
%N A286679 a(n) = (2*prime(n)^2 + 1)/3.
%C A286679 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 A286679 G. C. Greubel, <a href="/A286679/b286679.txt">Table of n, a(n) for n = 3..1000</a>
%F A286679 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 A286679 a(3) = 17; a(n + 1) = a(n) + 16 * A075888(n-2) for n > 3.
%F A286679 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 A286679 (2Prime[Range[3, 50]]^2 + 1)/3 (* _Alonso del Arte_, May 12 2017 *)
%o A286679 (PARI) {
%o A286679 forprime(n=5,300,
%o A286679 print1((2*n^2+1)/3", ")
%o A286679 )
%o A286679 }
%o A286679 (Magma) [(2*NthPrime(n)^2+1)/3: n in [3..50]]; // _Vincenzo Librandi_, May 15 2017
%Y A286679 Cf. A075888, A001318.
%K A286679 nonn,easy
%O A286679 3,1
%A A286679 _Dimitris Valianatos_, May 12 2017