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 A036468 #50 Apr 02 2017 10:24:33
%S A036468 1,2,2,2,3,3,4,4,4,3,4,8,4,6,5,4,9,8,6,9,7,7,7,5,7,9,14,8,9,11,7,17,
%T A036468 11,10,9,11,9,8,13,9,15,20,11,14,13,8,18,14,10,18,16,10,17,16,13,20,
%U A036468 20,13,14,17,12,23,18,14,22,15,17,18,21,12,19,29,16,23,21,14,27,24
%N A036468 Number of ways to represent 2n+1 as a+b with 0 < a < b and a^2 + b^2 prime.
%C A036468 Zhang Ming-Zhi (zamiz(AT)mail.sc.cninfo.net) asks if a(m) is always > 0.
%C A036468 I have confirmed that a(n) > 0 for 0 < n < 10^7. - _T. D. Noe_, Oct 17 2004
%C A036468 This open problem is mentioned by Guy at the end of section C1. - _T. D. Noe_, Apr 22 2009
%C A036468 a(n) <= phi(2n+1)/2, where phi(m) = A000010(m), while a(n) = phi(2n+1)/2 only for n = 1, 2, and 7. - _Thomas Ordowski_, Jan 25 2014
%C A036468 Records in a(n) are for 2n+1 = 3, 5, 11, 15, 25, 35, 55, 65, 85, 125, 145, 185, 205, 215, 235, 265, 295, 325, 365, 415, ... cf. A001750. - _Thomas Ordowski_, Mar 02 2017
%C A036468 a(n) tends to be larger for n == 2 (mod 5): see plot in Links. - _Robert Israel_, Mar 02 2017
%C A036468 Number of primes p = ((2n+1)^2 + x^2)/2 for positive integers x < 2n+1. - _Thomas Ordowski_, Mar 06 2017
%D A036468 R. K. Guy, Unsolved Problems in Theory of Numbers, Section C1.
%H A036468 T. D. Noe, <a href="/A036468/b036468.txt">Table of n, a(n) for n = 1..10000</a>
%H A036468 Gordon Hamilton, <a href="http://www.youtube.com/watch?v=aPs61cOaXHU">Unsolved K-12: Grade 7</a>, 2014. (video)
%H A036468 Robert Israel, <a href="/A036468/a036468.png">a(5k+j) for j=0,1,2,3,4</a>
%F A036468 a(n) = O(n/log(n)). - _Thomas Ordowski_, Feb 11 2013
%p A036468 a:= n-> add(`if`(isprime(i^2+(2*n+1-i)^2), 1, 0), i=1..n):
%p A036468 seq(a(n), n=1..80); # _Alois P. Heinz_, Jul 09 2016
%t A036468 Table[cnt=0; m=2n+1; Do[If[PrimeQ[k^2+(m-k)^2], cnt++ ], {k, n}]; cnt, {n, 100}]
%o A036468 (PARI) a(n)=sum(k=1,n,isprime(k^2+(2*n-k+1)^2)) \\ _Charles R Greathouse IV_, Jan 09 2014
%Y A036468 Cf. A082519, A099468, A281543.
%K A036468 nonn
%O A036468 1,2
%A A036468 _R. K. Guy_
%E A036468 More terms from _David W. Wilson_ and _Michael Kleber_