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A267826 Numbers not of the form w^3 + 2*x^3 + 3*y^3 + 4*z^3, where w, x, y and z are nonnegative integers. 3
18, 22, 39, 60, 63, 74, 76, 77, 100, 103, 106, 107, 117, 126, 178, 180, 201, 215, 228, 230, 245, 271, 289, 291, 295, 315, 341, 356, 357, 393, 413, 419, 420, 480, 481, 523, 559, 606, 616, 671, 673, 705, 854, 855, 963, 980, 981, 998, 1103, 1121, 1130, 1298, 1484, 1510, 1643, 1729, 1849, 1916, 1934, 1946 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Conjecture: The sequence has exactly 122 terms the last of   which is a(122) = 41405.

We have verified that there are no terms between 41406 and 2*10^5.

The conjecture implies that {P(v)+w^3+2*x^3+3*y^3+4*z^3: w,x,y,z = 0,1,2,...} = {0,1,2,...} whenever P(v) is among the polynomials a*v^3 (a = 1,5,6,7,9,10,12,15,18), b*v^4 (b = 1,2,3,5,6,12,18), c*v^5 (c = 1,2,5,12) and d*v^k (d = 5,12; k = 6,7). Moreover, it also implies that {8*t+w^3+2*x^3+3*y^3+4*z^3: t = 0,1; w,x,y,z = 0,1,2,...} = {0,1,2,...}. If a,b,c,d and m are positive integers with {m*t+a*w^3+b*x^3+c*y^3+d*z^3: t = 0,1; w,x,y,z = 0,1,2,...} = {0,1,2,...}, then we must have m = 8 and {a,b,c,d} = {1,2,3,4}.

LINKS

Zhi-Wei Sun, Table of n, a(n) for n = 1..122

Zhi-Wei Sun, Universal sums u^3+a*v^3+b*x^3+c*y^3+d*z^3 with u, v, x, y, z nonnegative integers, a message to Number Theory Mailing List, April 3, 2016.

EXAMPLE

a(1) = 18 since it is the first nonnegative integer not in the set {w^3 + 2*x^3 + 3*y^3 + 4*z^3: w,x,y,z = 0,1,2,...}.

MATHEMATICA

CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]

n=0; Do[Do[If[CQ[m-4*z^3-3y^3-2x^3], Goto[aa]], {z, 0, (m/4)^(1/3)}, {y, 0, ((m-4z^3)/3)^(1/3)}, {x, 0, ((m-4z^3-3y^3)/2)^(1/3)}]; n=n+1; Print[n, " ", m]; Label[aa]; Continue, {m, 0, 1946}]

CROSSREFS

Cf. A000578, A002804, A022566, A267861, A271099, A271169, A271237.

Sequence in context: A250737 A290172 A031407 * A339473 A002505 A182438

Adjacent sequences:  A267823 A267824 A267825 * A267827 A267828 A267829

KEYWORD

nonn

AUTHOR

Zhi-Wei Sun, Apr 07 2016

STATUS

approved

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Last modified August 1 07:45 EDT 2021. Contains 346384 sequences. (Running on oeis4.)