A262959 Positive integers that cannot be written as the sum of a fourth power, a square and a positive triangular number.

9, 34, 63, 89, 99, 139, 164, 174, 193, 204, 245, 314, 324, 399, 424, 454, 464, 489, 504, 524, 549, 714, 1049, 1089, 1149, 1174, 1439, 1504, 1539, 1639, 1714, 1799, 1814, 1919, 2164, 2239, 2313, 2374, 2414, 2439, 2764, 2789, 3079, 3319, 3414, 3669, 3774, 3814, 4019, 4114

Offset

1,1

Comments

Conjecture: (i) Each term is congruent to one of 3, 4, 5, 9 modulo 10.

(ii) a(n+1) - a(n) > 4 for all n > 0.

Part (ii) of this conjecture is stronger than the conjecture in A262956. Note that a(139) - a(138) = 18089 - 18084 = 5.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..500

Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.

Example

a(1) = 9 since each of 1..8 can be written as x^4 + y^2 + z*(z+1)/2 with z > 0, but 9 cannot be represented in this way. Clearly, 1 = 0^4 + 0^2 + 1*2/2, 2 = 0^4 + 1^2 + 1*2/2, 3 = 1^4 + 1^2 + 1*2/2, 4 = 0^4 + 1^2 + 2*3/2, 5 = 1^4 + 1^2 + 2*3/2, 6 = 0^4 + 0^2 + 3*4/2, 7 = 0^4 + 1^2 + 3*4/2 and 8 = 1^3 + 1^2 + 3*4/2.

Mathematica

TQ[n_]:=TQ[n]=n>0&&IntegerQ[Sqrt[8n+1]]
n=0; Do[Do[If[TQ[m-x^4-y^2], Goto[aa]], {x, 0, m^(1/4)}, {y, 0, Sqrt[m-x^4]}]; n=n+1; Print[n, " ", m]; Label[aa]; Continue, {m, 1, 5000}]

Crossrefs

Cf. A000217, A000290, A000583, A262956.

Sequence in context: A044086 A044467 A020163 * A139275 A236370 A273744

Adjacent sequences: A262956 A262957 A262958 * A262960 A262961 A262962

Keyword

nonn

Author

Zhi-Wei Sun, Oct 05 2015

Status

approved