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 A301534 #22 Apr 16 2018 08:09:02
%S A301534 0,2,3,4,5,5,2,6,6,4,7,4,9,6,6,6,7,9,5,10,3,9,7,9,8,11,9,8,10,5,8,9,4,
%T A301534 10,7,7,7,8,7,13,8,6,6,14,7,15,3,11,8,10,8,7,7,9,6,9,7,7,10,12,6,9,4,
%U A301534 7,10,12,12,7,13,9,12,6,7,10,5,8,7,12,12,10
%N A301534 Number of ways to write the n-th prime congruent to 7 modulo 12 as x^2 + 3*y^2 + 15*2^z with x,y,z nonnegative integers.
%C A301534 Conjecture: a(n) > 0 for all n > 1. In other words, any prime p > 7 with p == 7 (mod 12) can be written as x^2 + 3*y^2 + 15*2^z with x,y,z nonnegative integers.
%C A301534 We have verified the conjecture for all primes p == 7 (mod 12) with 7 < p < 8*10^9.
%H A301534 Zhi-Wei Sun, <a href="/A301534/b301534.txt">Table of n, a(n) for n = 1..10000</a>
%e A301534 a(1) = 0 since 7 cannot be written as x^2 + 3*y^2 + 15*2^z with x,y,z nonnegative integers.
%e A301534 a(2) = 2 since the second prime congruent to 7 modulo 12 is 19 and 19 = 1^2 + 3*1^2 + 15*2^0 = 2^2 + 3*0^2 + 15*2^0.
%t A301534 p[n_]:=p[n]=Prime[n];
%t A301534 SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
%t A301534 f[n_]:=f[n]=FactorInteger[n];
%t A301534 g[n_]:=g[n]=Sum[Boole[MemberQ[{2},Mod[Part[Part[f[n],i],1],3]]&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
%t A301534 QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);
%t A301534 n=0;Do[If[Mod[p[m],12]!=7,Goto[aa]];n=n+1;r=0;Do[If[QQ[p[m]-15*2^k],Do[If[SQ[p[m]-15*2^k-3x^2],r=r+1],{x,0,Sqrt[(p[m]-15*2^k)/3]}]],{k,0,Log[2,p[m]/15]}];Print[n," ",r];Label[aa],{m,1,315}]
%Y A301534 Cf. A000040, A000079, A000290, A092572, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301472, A302920.
%K A301534 nonn
%O A301534 1,2
%A A301534 _Zhi-Wei Sun_, Apr 16 2018