A301534 - OEIS

%I #22 Apr 16 2018 08:09:02

%S 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 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 7,10,12,12,7,13,9,12,6,7,10,5,8,7,12,12,10

%N 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 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 We have verified the conjecture for all primes p == 7 (mod 12) with 7 < p < 8*10^9.

%H Zhi-Wei Sun, <a href="/A301534/b301534.txt">Table of n, a(n) for n = 1..10000</a>

%e a(1) = 0 since 7 cannot be written as x^2 + 3*y^2 + 15*2^z with x,y,z nonnegative integers.

%e 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 p[n_]:=p[n]=Prime[n];

%t SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];

%t f[n_]:=f[n]=FactorInteger[n];

%t 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 QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);

%t 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 Cf. A000040, A000079, A000290, A092572, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301472, A302920.

%K nonn

%O 1,2

%A _Zhi-Wei Sun_, Apr 16 2018