OFFSET
1,2
COMMENTS
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.
We have verified the conjecture for all primes p == 7 (mod 12) with 7 < p < 8*10^9.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
EXAMPLE
a(1) = 0 since 7 cannot be written as x^2 + 3*y^2 + 15*2^z with x,y,z nonnegative integers.
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.
MATHEMATICA
p[n_]:=p[n]=Prime[n];
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
f[n_]:=f[n]=FactorInteger[n];
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;
QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);
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}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Apr 16 2018
STATUS
approved