

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
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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 padic order of n is even for any prime p == 5 or 7 (mod 8).


LINKS

ZhiWei Sun, Table of n, a(n) for n = 1..10000


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]==5Mod[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^33*2^k], Goto[aa]], {k, 0, Log[2, m^3/3]}]; tab=Append[tab, m]; Label[aa], {m, 1, 429}]; Print[tab]


CROSSREFS

Cf. A000079, A000578, A002479, A301471, A301472.
Sequence in context: A160029 A229663 A223091 * A244187 A045807 A007644
Adjacent sequences: A301476 A301477 A301478 * A301480 A301481 A301482


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 22 2018


STATUS

approved



