|
|
|
|
|
|
|
|
|
#59 by Charles R Greathouse IV at Sun Aug 04 14:11:00 EDT 2024
|
|
| REFERENCES
|
B. Benfield and O. Lippard, <a href
arXiv:2404.08190>End Behavior of Ramanujan's Taxicab Numbers</a>
|
|
| LINKS
|
B. Benfield and O. Lippard, <a href="https://arxiv.org/abs/2404.08190">End Behavior of Ramanujan's Taxicab Numbers</a>
|
|
|
|
|
|
|
#58 by Charles R Greathouse IV at Sun Aug 04 13:52:24 EDT 2024
|
|
| 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) = 17.5603exp(-0.0327825n).
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) = 17.5603exp(-0.0327825n).
|
|
| EXAMPLE
|
Example: For k = 3, Taxicab(2,j,3) does not exist for all j > 9, hence 3 is not a member of the sequence.
|
|
|
|
|
Discussion
|
| Sun Aug 04
| 13:54
| Charles R Greathouse IV: I hope I’m not misunderstanding the example?
|
|
| 13:56
| Charles R Greathouse IV: I don’t understand the second conjecture, doesn’t that go to 0 at infinity? For exponential growth I’d expect a positive exponent.
|
|
| 13:56
| Charles R Greathouse IV: The sequence is definitely interesting, I just want to get it cleaned up before publication.
|
|
|
|
|
|
|
|
|
|
|
|
|
#62 by Charles R Greathouse IV at Fri Aug 02 22:39:38 EDT 2024
|
|
|
|
|
|
|
|
|
|
|
|
|
|
#253 by Charles R Greathouse IV at Fri Aug 02 22:39:27 EDT 2024
|
|
|
|
|
|
|
|
|
|
|
|
|
|
#37 by Charles R Greathouse IV at Fri Aug 02 22:39:16 EDT 2024
|
|
|
|
|
|
|
|
|
|
|
|
|
|
#116 by Charles R Greathouse IV at Fri Aug 02 22:39:10 EDT 2024
|
|
|
|
|
|
|
|
|
| A006353
|
|
Expansion of (phi(-q^3) * psi(q))^3 / (phi(-q) * psi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
(history;
published version)
|
|
|
|
|
#73 by Charles R Greathouse IV at Fri Aug 02 22:38:58 EDT 2024
|
|
|
|
|
|
|
|
|
|
|
|
|
|
#76 by Charles R Greathouse IV at Fri Aug 02 22:38:45 EDT 2024
|
|
|
|
|
|
|
|
|
|
|
|
|
|
#32 by Charles R Greathouse IV at Fri Aug 02 22:38:39 EDT 2024
|
|
|
|
|
|
|
|
|
|
|
|
|
|
#31 by Charles R Greathouse IV at Fri Aug 02 22:38:29 EDT 2024
|
|
|
|
|
|
|