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A374486
Numbers k such that Taxicab(2,j,k) exists for large j.
1
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
1,2
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
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
STATUS
approved