OFFSET
1,1
COMMENTS
If x^3 = y^4 + z^2, then (a^(4k)*x)^3 = (a^(3k)*y)^4 + (a^(6k)*z)^2 for all a = 1,2,3,... and k = 0,1,2,... So the sequence has infinitely many terms.
Conjecture: For any integer m, there are infinitely many triples (x,y,z) of positive integers with x^4 - y^3 + z^2 = m.
This is stronger than the conjecture in A266152.
LINKS
Zhi-Wei Sun and Chai Wah Wu, Table of n, a(n) for n = 1..698 n = 1..100 from Zhi-Wei Sun
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
EXAMPLE
a(1) = 8 since 8^3 = 4^4 + 16^2.
a(2) = 13 since 13^3 = 3^4 + 46^2.
a(3) = 20 since 20^3 = 4^4 + 88^2.
a(8) = 193 since 193^3 = 6^4 + 2681^2.
a(12) = 313 since 313^3 = 66^4 + 3419^2.
a(20) = 1000 since 1000^3 = 100^4 + 30000^2.
MATHEMATICA
SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
n=0; Do[Do[If[SQ[x^3-y^4], n=n+1; Print[n, " ", x]; Goto[aa]], {y, 1, x^(3/4)}]; Label[aa]; Continue, {x, 1, 8000}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 23 2015
STATUS
approved