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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 (list; graph; refs; listen; history; text; internal format)
 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). LINKS Zhi-Wei 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]==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 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 Zhi-Wei Sun, Mar 22 2018 STATUS approved

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Last modified July 16 13:33 EDT 2020. Contains 335788 sequences. (Running on oeis4.)