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A301479
Positive integers m such that m^3 cannot be written in the form x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers.
7
1, 53, 69, 71, 77, 87, 101, 103, 106, 117, 127, 133, 138, 142, 149, 159, 173, 174, 181, 191, 197, 199, 202, 206, 207, 212, 213, 221, 223, 229, 231, 234, 266, 269, 276, 277, 284, 293, 298, 309, 311, 325, 341, 346, 348, 351, 357, 362, 365, 373, 389, 398, 404, 412, 423, 424, 426, 429
OFFSET
1,2
COMMENTS
It seems that this sequence has infinitely many terms. In contrast, the author conjectured in A301471 and A301472 that any square greater than one can be written as x^2 + 2*y^2 + 3*2^z with x,y,z nonnegative integers.
It is known that a positive integer n has the form x^2 + 2*y^2 with x and y integers if and only if the p-adic order of n is even for any prime p == 5 or 7 (mod 8).
EXAMPLE
a(1) = 1 since x^2 + 2*y^2 + 3*2^z > 1^3 for all x,y,z = 0,1,2,....
MATHEMATICA
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]);
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={}; Do[Do[If[QQ[m^3-3*2^k], Goto[aa]], {k, 0, Log[2, m^3/3]}]; tab=Append[tab, m]; Label[aa], {m, 1, 429}]; Print[tab]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Mar 22 2018
STATUS
approved