

A270921


Number of ordered ways to write n as x*(3x+2) + y*(5y+1)/2  z^4, where x and y are integers, and z is a nonnegative integer with z^4 <= n.


2



1, 2, 3, 3, 2, 1, 1, 3, 3, 3, 4, 3, 1, 1, 1, 2, 5, 3, 3, 3, 3, 4, 3, 4, 6, 3, 6, 4, 3, 4, 2, 3, 3, 2, 2, 3, 2, 2, 4, 3, 3, 5, 9, 6, 3, 4, 2, 2, 2, 6, 3, 3, 2, 2, 3, 2, 2, 4, 5, 4, 5, 3, 2, 2, 6, 7, 4, 4, 2, 2, 4, 3, 3, 3, 2, 3, 4, 4, 4, 2, 5
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OFFSET

0,2


COMMENTS

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 0, 5, 6, 12, 13, 14, 112, 193, 194, 200, 242, 333, 345, 376, 492, 528, 550, 551, 613, 797, 1178, 1195, 1222, 1663, 3380, 3635, 6508, 8755, 9132, 12434, 20087.
Compare this conjecture with the conjecture in A270566.


LINKS

ZhiWei Sun, Table of n, a(n) for n = 0..10000
Z.W. Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103113.
Z.W. Sun, On universal sums of polygonal numbers, Sci. China Math. 58(2015), no. 7, 13671396.
Z.W. Sun, On universal sums ax^2+by^2+f(z), aT_x+bT_y+f(z) and zT_x+by^2+f(z), preprint, arXiv:1502.03056 [math.NT], 2015.


EXAMPLE

a(5) = 1 since 5 = 1*(3*1+2) + 0*(5*0+1)/2  0^4.
a(6) = 1 since 6 = 1*(3*1+2) + (1)*(5*(1)+1)/2  1^4.
a(13) = 1 since 13 = 1*(3*1+2) + (2)*(5*(2)+1)/2  1^4.
a(376) = 1 since 376 = 0*(3*0+2) + (16)*(5*(16)+1)/2  4^4.
a(9132) = 1 since 9132 = (13)*(3*(13)+2) + 59*(5*59+1)/2  3^4.
a(12434) = 1 since 12434 = (21)*(3*(21)+2) + 78*(5*78+1)/2  8^4.
a(20087) = 1 since 20087 = 19*(3*19+2) + 87*(5*87+1)/2  0^4.
5, 6, 12, 13, 14, 112, 193, 194, 200, 242, 333, 345, 376, 492, 528, 550, 551, 613, 797, 1178, 1195, 1222, 1663, 3380, 3635, 6508, 8755, 9132, 12434, 20087


MATHEMATICA

pQ[n_]:=pQ[n]=IntegerQ[Sqrt[40n+1]]&&(Mod[Sqrt[40n+1], 10]==1Mod[Sqrt[40n+1], 10]==9)
Do[r=0; Do[If[pQ[n+x^4y(3y+2)], r=r+1], {x, 0, n^(1/4)}, {y, Floor[(Sqrt[3(n+x^4)+1]+1)/3], (Sqrt[3(n+x^4)+1]1)/3}]; Print[n, " ", r]; Continue, {n, 0, 80}]


CROSSREFS

Cf. A000583, A001082, A262813, A262815, A262816, A262827, A270469, A270488, A270516, A270533, A270559, A270566, A270920.
Sequence in context: A274885 A287732 A171414 * A038529 A176259 A132312
Adjacent sequences: A270918 A270919 A270920 * A270922 A270923 A270924


KEYWORD

nonn


AUTHOR

ZhiWei Sun, Mar 25 2016


STATUS

approved



