OFFSET
1,1
COMMENTS
Conjecture: (i) Each term is congruent to one of 3, 4, 5, 9 modulo 10.
(ii) a(n+1) - a(n) > 4 for all n > 0.
Part (ii) of this conjecture is stronger than the conjecture in A262956. Note that a(139) - a(138) = 18089 - 18084 = 5.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..500
Zhi-Wei Sun, Mixed sums of squares and triangular numbers, Acta Arith. 127(2007), 103-113.
EXAMPLE
a(1) = 9 since each of 1..8 can be written as x^4 + y^2 + z*(z+1)/2 with z > 0, but 9 cannot be represented in this way. Clearly, 1 = 0^4 + 0^2 + 1*2/2, 2 = 0^4 + 1^2 + 1*2/2, 3 = 1^4 + 1^2 + 1*2/2, 4 = 0^4 + 1^2 + 2*3/2, 5 = 1^4 + 1^2 + 2*3/2, 6 = 0^4 + 0^2 + 3*4/2, 7 = 0^4 + 1^2 + 3*4/2 and 8 = 1^3 + 1^2 + 3*4/2.
MATHEMATICA
TQ[n_]:=TQ[n]=n>0&&IntegerQ[Sqrt[8n+1]]
n=0; Do[Do[If[TQ[m-x^4-y^2], Goto[aa]], {x, 0, m^(1/4)}, {y, 0, Sqrt[m-x^4]}]; n=n+1; Print[n, " ", m]; Label[aa]; Continue, {m, 1, 5000}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 05 2015
STATUS
approved