cp's OEIS Frontend

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

%I A309290 #56 Mar 10 2025 16:39:34
%S A309290 0,2,5,7,11,17,19,23,29,31,33,35,41,43,47,59,61,65,67,71,73,77,79,83,
%T A309290 89
%N A309290 Numbers k such that binomial(k^2,k) - k^2 is squarefree.
%C A309290 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]
%C A309290 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
%C A309290 113 is a term. - _Chai Wah Wu_, Jul 20 2020
%C A309290 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
%H A309290 Juri-Stepan Gerasimov and others, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2018-March/018478.html">C(2n,n) - n^2 and C(n^2,n) - n^n</a>, SeqFan list, April 20, 2018.
%t A309290 Select[Range[0, 50], SquareFreeQ[Binomial[#^2, #] - #^2] &] (* _Vincenzo Librandi_, Jul 31 2019 *)
%o A309290 (PARI) is(n)=issquarefree(binomial(n^2,n)-n^2)
%o A309290 for(n=0,oo, is(n) && print1(n,", "))
%o A309290 (Magma) [0] cat  [n: n in [2..45] | IsSquarefree(Binomial(n^2, n) - n^2)]; // _Vincenzo Librandi_, Jul 31 2019
%Y A309290 Cf. A308078 (binomial(n^2,n) - n^n is squarefree), A309289 (binomial(2n,n) - n^2 is prime).
%K A309290 nonn,hard,more
%O A309290 1,2
%A A309290 _M. F. Hasler_ (at the suggestion of _Juri-Stepan Gerasimov_ and others), Jul 21 2019
%E A309290 a(24) from _Daniel Suteu_, Jul 30 2019
%E A309290 a(25) from _Tyler Busby_, Mar 10 2025