A309290 Numbers k such that binomial(k^2,k) - k^2 is squarefree.

0, 2, 5, 7, 11, 17, 19, 23, 29, 31, 33, 35, 41, 43, 47, 59, 61, 65, 67, 71, 73, 77, 79, 83

Offset

1,2

Comments

Sequence probably continues ..., 89, 97, ... [There is no composite term between 77 and 161 except possibly 133, 143 and 145. - M. F. Hasler, Feb 15 2022]

The sequence appears to contain most primes (except 3, 13, 37, 53, ...) and some semiprimes (33, 65, 77, ...). What can be said about these "exceptional" values? What are the first terms with more prime factors? The sequence remains nearly the same if k^2 is replaced by k^k. (Then 0 and 11 are not in the sequence but 3, 13, 37 and 53 are.) - M. F. Hasler, Jul 31 2019

113 is a term. - Chai Wah Wu, Jul 20 2020

Links

Table of n, a(n) for n=1..24.

Juri-Stepan Gerasimov and others, C(2n,n) - n^2 and C(n^2,n) - n^n, SeqFan list, April 20, 2018.

Mathematica

Select[Range[0, 50], SquareFreeQ[Binomial[#^2, #] - #^2] &] (* Vincenzo Librandi, Jul 31 2019 *)

Prog

(PARI) is(n)=issquarefree(binomial(n^2, n)-n^2)
for(n=0, oo, is(n) && print1(n, ", "))
(Magma) [0] cat  [n: n in [2..45] | IsSquarefree(Binomial(n^2, n) - n^2)]; // Vincenzo Librandi, Jul 31 2019

Crossrefs

Cf. A308078 (binomial(n^2,n) - n^n is squarefree), A309289 (binomial(2n,n) - n^2 is prime).

Sequence in context: A342939 A089781 A144832 * A023238 A194991 A345667

Adjacent sequences: A309287 A309288 A309289 * A309291 A309292 A309293

Keyword

nonn,hard,more

Author

M. F. Hasler (at the suggestion of Juri-Stepan Gerasimov and others), Jul 21 2019

Extensions

a(24) from Daniel Suteu, Jul 30 2019

Status

approved