A308078 -id:A308078 - cp's OEIS frontend

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, 89

Offset: 1

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

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
Since binomial(k^2,k) = k*binomial(k^2-1,k-1), each term k is squarefree and coprime to binomial(k^2-1,k-1). It follows that the smallest candidate term with at least 3 prime factors is k = 935. - Max Alekseyev, Mar 04 2025

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

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

[0] cat  [n: n in [2..45] | IsSquarefree(Binomial(n^2, n) - n^2)]; // Vincenzo Librandi, Jul 31 2019

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

is(n)=issquarefree(binomial(n^2,n)-n^2)
for(n=0,oo, is(n) && print1(n,", "))

a(24) from Daniel Suteu, Jul 30 2019

a(25) from Tyler Busby, Mar 10 2025

cp's OEIS Frontend

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

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