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A302920
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Number of ways to write prime(n)^2 as x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers.
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15
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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, 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, 7, 15, 5, 11, 8, 14, 10, 7, 10
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OFFSET
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1,2
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COMMENTS
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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.
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.
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LINKS
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EXAMPLE
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a(1) = 1 with prime(1)^2 = 4 = 1^2 + 2*0^2 + 3*2^0.
a(2) = 2 with prime(2)^2 = 9 = 2^2 + 2*1^2 + 3*2^0 = 1^2 + 2*1^2 + 3*2^1.
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MATHEMATICA
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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[(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;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
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]
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CROSSREFS
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Cf. A000040, A000079, A000290, A002479, A299924, A299537, A299794, A300219, A300362, A300396, A300510, A301376, A301391, A301452, A301471, A301472.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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