OFFSET
1,2
COMMENTS
Here Taxicab(2,j,k) denotes the smallest number (if it exists) that is the sum of j perfect squares in exactly k ways. For sufficiently large N, Taxicab(2,j,k) either always exists for j > N or always does not exist for j > N.
Conjecture: Infinitely many positive integers are in this sequence, and infinitely many positive integers are not in this sequence.
Conjecture: This sequence grows exponentially. Computationally it appears to have asymptotic a(n) = 1.03691*exp(0.594473*n^(1/2)).
REFERENCES
E. Grosswald. Representations of Integers as Sums of Squares. Springer New York, NY, 1985.
LINKS
Oliver Lippard, Table of n, a(n) for n = 1..372
B. Benfield, O. Lippard, and A. Roy, End Behavior of Ramanujan's Taxicab Numbers, arXiv:2404.08190 [math.NT], 2024.
EXAMPLE
For k = 3, Taxicab(2,j,3) does not exist for all j > 9, hence 3 is not a member of the sequence.
CROSSREFS
KEYWORD
nonn
AUTHOR
Oliver Lippard and Brennan G. Benfield, Aug 04 2024
STATUS
approved