cp's OEIS Frontend

A308078 Numbers k such that binomial(k^2,k) - k^k is squarefree.

%I A308078 #49 Apr 21 2025 03:30:54
%S A308078 2,3,5,7,13,17,19,23,29,31,33,35,37,41,43,47,53,59,61,65,67,71,73,77,
%T A308078 79,83,85,89,97
%N A308078 Numbers k such that binomial(k^2,k) - k^k is squarefree.
%C A308078 The sequence appears to contain most primes (except 11, ...) and some odd semiprimes (33, 65, 77, ...). What can be said about these "exceptional" values? What are the first terms with more prime factors?
%C A308078 The sequence remains nearly the same if k^k is replaced by k^2, cf. A309290. Then 0 and 11 are in the sequence but 3, 13, 37 and 53 are not.
%C A308078 97 is a term (see SeqFan list discussion). - _Chai Wah Wu_, Jul 20 2020
%C A308078 Similarly to 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 05 2025
%H A308078 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 A308078 Select[Range[0, 50], SquareFreeQ[Binomial[#^2, #] - #^#] &]
%o A308078 (PARI) is(n)=issquarefree(binomial(n^2,n)-n^n)
%o A308078 for(n=0,oo, is(n) && print1(n,", "))
%o A308078 (Magma) [0] cat  [n: n in [2..45] | IsSquarefree(Binomial(n^2, n) - n^n)];
%Y A308078 Cf. A309289 (binomial(2k,k) - k^2 is prime), A309290 (binomial(k^2,k) - k^2 is squarefree).
%K A308078 nonn,hard,more
%O A308078 1,1
%A A308078 _M. F. Hasler_ (at the suggestion of _Juri-Stepan Gerasimov_ and others), Jul 31 2019
%E A308078 a(26) from _Max Alekseyev_, Mar 09 2025
%E A308078 a(27) from _Tyler Busby_, Mar 10 2025
%E A308078 a(28)-a(29) from _Tyler Busby_, Apr 16 2025