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%I #11 Apr 16 2018 08:09:09
%S 1,2,3,3,4,5,4,4,3,7,6,7,6,7,8,8,7,7,6,5,7,6,8,6,8,7,9,9,7,6,6,9,7,5,
%T 8,5,9,9,10,10,9,14,7,5,11,8,8,11,10,10,12,10,6,12,11,10,8,9,10,11,8,
%U 7,15,5,11,8,14,10,7,10
%N Number of ways to write prime(n)^2 as x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers.
%C Conjecture: a(n) > 0 for all n > 0. In other words, for any prime p there are nonnegative integers x, y and z such that x^2 + 2*y^2 + 3*2^z = p^2.
%C As mentioned in A301471, for the composite number m = 5884015571 = 7*17*49445509 there are no nonnegative integers x,y,z such that x^2 + 2*y^2 + 3*2^z = m^2.
%H Zhi-Wei Sun, <a href="/A302920/b302920.txt">Table of n, a(n) for n = 1..6000</a>
%H Zhi-Wei Sun, <a href="http://dx.doi.org/10.1016/j.jnt.2016.11.008">Refining Lagrange's four-square theorem</a>, J. Number Theory 175(2017), 167-190.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1701.05868">Restricted sums of four squares</a>, arXiv:1701.05868 [math.NT], 2017-2018.
%e a(1) = 1 with prime(1)^2 = 4 = 1^2 + 2*0^2 + 3*2^0.
%e a(2) = 2 with prime(2)^2 = 9 = 2^2 + 2*1^2 + 3*2^0 = 1^2 + 2*1^2 + 3*2^1.
%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[(Mod[Part[Part[f[n],i],1],8]==5||Mod[Part[Part[f[n],i],1],8]==7)&&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 tab={};Do[r=0;Do[If[QQ[p[n]^2-3*2^k],Do[If[SQ[p[n]^2-3*2^k-2x^2],r=r+1],{x,0,Sqrt[(p[n]^2-3*2^k)/2]}]],{k,0,Log[2,p[n]^2/3]}];tab=Append[tab,r],{n,1,70}];Print[tab]
%Y Cf. A000040, A000079, A000290, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301472.
%K nonn
%O 1,2
%A _Zhi-Wei Sun_, Apr 15 2018
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