

A334086


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


3



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
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OFFSET

1,1


COMMENTS

Conjecture: The sequence has totally 216 terms as listed in the bfile.
As none of the 216 terms in the bfile 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 + (yz)^2, where x,y,z are integers.
Our computation indicates that after the 216th 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

ZhiWei Sun, Table of n, a(n) for n = 1..216
ZhiWei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103113.
ZhiWei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34 (2017), no. 2, 97120. (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[n2x^4y(y+1)/2], Goto[aa]], {x, 0, (n/2)^(1/4)}, {y, 0, (Sqrt[4(n2x^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

ZhiWei Sun, Apr 14 2020


STATUS

approved



