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

1,1

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

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).)

Example

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.

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]

Crossrefs

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

Sequence in context: A212056 A290224 A071633 * A217052 A044206 A044587

Adjacent sequences: A334083 A334084 A334085 * A334087 A334088 A334089

Keyword

nonn

Author

Zhi-Wei Sun, Apr 14 2020

Status

approved