OFFSET
1,2
COMMENTS
Conjecture: a(n) > 0 for all n > 0. In other words, for any positive integer n, either n or n + 1 can be written as the sum of a fourth power, a square and a positive triangular number.
We also guess that a(n) = 1 only for n = 1, 89, 244, 464, 5243, 14343.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
EXAMPLE
a(1) = 1 since 1 = 0^4 + 1*2/2 + 0^2.
a(89) = 1 since 89 = 2^4 + 4*5/2 + 8^2 - 1.
a(244) = 1 since 244 = 2^4 + 2*3/2 + 15^2.
a(464) = 1 since 464 = 2^4 + 22*23/2 + 14^2 - 1.
a(5243) = 1 since 5243 = 0^4 + 50*51/2 + 63^2 - 1.
a(14343) = 1 since 14343 = 2^4 + 163*164/2 + 31^2.
MATHEMATICA
SQ[n_]:=IntegerQ[Sqrt[n]]||IntegerQ[Sqrt[n+1]]
Do[r=0; Do[If[SQ[n-x^4-y(y+1)/2], r=r+1], {x, 0, n^(1/4)}, {y, 1, (Sqrt[8(n-x^4)+1]-1)/2}]; Print[n, " ", r]; Continue, {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 05 2015
STATUS
approved