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

Author

Zhi-Wei Sun, Oct 05 2015

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A000217, A000290, A000583, A262956.

Programs

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}]