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 A194830 #10 Feb 16 2025 08:33:15
%S A194830 7,9,17,33,45,51,75,93,105,135,153,165,225,273,285,315,405,465,495,
%T A194830 525,735,765,945,1155,1365,1785,1995,2145,2415,2625,3045,3255,3465,
%U A194830 3885,4095,4305,4725,4935,5145,5355,5565,5775,6405,6825,7665,8085,8925,9555
%N A194830 Odd numbers for which the number of ways to express them as 2*p+q, p,q primes, is a new record.
%C A194830 2*(positions of records in A046927)+1. The corresponding new record number of representations is given in A194831.
%D A194830 See A046927.
%H A194830 Hugo Pfoertner, <a href="/A194830/b194830.txt">Table of n, a(n) for n = 1..322</a>
%H A194830 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LevysConjecture.html">Levy's Conjecture</a>
%e A194830 a(3)=17, because it can be represented in A194831(3)=4 different ways (17=2*2+13=2*3+11=2*5+7=2*7+3), whereas all smaller odd numbers can only be represented in a smaller number of ways.
%t A194830 ways[n_] := ways[n] = (w = 0; Do[ p = 2k + 1; q = n - k; If[PrimeQ[p] && PrimeQ[q], w++], {k, 1, n}]; w); record = 0; A194830 = Reap[Do[If[ways[n] > record, record = ways[n]; Print["2n+1 = ", 2n + 1, " record = ", record]; Sow[{ways[n], 2n + 1}]], {n, 0, 12000}]][[2, 1]][[All, 2]] (* _Jean-François Alcover_, Dec 05 2012 *)
%Y A194830 Cf. A046927, A194831 (size of records).
%K A194830 nonn
%O A194830 1,1
%A A194830 _Hugo Pfoertner_, Sep 12 2011