A334086 Positive numbers not of the form 2*x^4 + y*(y+1)/2 + z*(z+1)/2 with x,y,z nonnegative integers.

19, 82, 109, 118, 145, 149, 271, 280, 296, 349, 350, 371, 392, 454, 491, 643, 670, 692, 754, 755, 923, 937, 986, 989, 1021, 1031, 1150, 1189, 1210, 1294, 1346, 1372, 1610, 1682, 1699, 1720, 1819, 1913, 2050, 2065, 2141, 2227, 2479, 2524, 2753, 2996, 3184, 3451, 3590, 3805, 3968, 4129, 4139, 4199, 4261, 4706

Offset: 1

Author

Zhi-Wei Sun, Apr 14 2020

Keywords

• nonn

Comments

Conjecture: The sequence has totally 216 terms as listed in the b-file.

As none of the 216 terms in the b-file is divisible by 3, the conjecture implies that for each nonnegative integer n we can write 3*n as 2*x^4 + y*(y+1)/2 + z*(z+1)/2 and hence 12*n+1 = 8*x^4 + (y+z+1)^2 + (y-z)^2, where x,y,z are integers.

Our computation indicates that after the 216-th term 4592329 there are no other terms below 10^8.

It is known that each n = 0,1,2,... can be written as the sum of two triangular numbers and twice a square.

a(217) > 10^9, if it exists. - Giovanni Resta, Apr 14 2020

Examples

			a(1) = 19 since 19 is the first nonnegative integer which cannot be written as the sum of two triangular numbers and twice a fourth power.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..216
• Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
• Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no. 2, 97-120. (Cf. Conjecture 1.4(ii).)

Crossrefs

Cf. A000217, A000583, A115160, A290491, A306227, A334113.

Programs

Mathematica

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