A374486 Numbers k such that Taxicab(2,j,k) exists for large j.

1, 2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 33, 37, 39, 40, 42, 44, 48, 50, 51, 52, 53, 56, 59, 62, 66, 68, 70, 72, 74, 77, 79, 87, 91, 92, 96, 97, 103, 108, 112, 115, 117, 120, 121, 124, 130, 131, 138, 148, 149, 161, 164, 176, 184, 185, 194, 200

Offset: 1

Author

Oliver Lippard and Brennan G. Benfield, Aug 04 2024

Keywords

• nonn

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)).

Examples

			For k = 3, Taxicab(2,j,3) does not exist for all j > 9, hence 3 is not a member of the sequence.
		

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.

Crossrefs

Cf. A025416, A080673, A295702, A295795