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%I #36 Jan 02 2023 12:30:54
%S 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 79
%N Numbers k such that binomial(k^2,k) - k^k is squarefree.
%C 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 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 97 is a term (see SeqFan list discussion). - _Chai Wah Wu_, Jul 20 2020
%H Juri-Stepan Gerasimov and others, <a href="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 Select[Range[0, 50], SquareFreeQ[Binomial[#^2, #] - #^#] &]
%o (PARI) is(n)=issquarefree(binomial(n^2,n)-n^n)
%o for(n=0,oo, is(n) && print1(n,", "))
%o (Magma) [0] cat [n: n in [2..45] | IsSquarefree(Binomial(n^2, n) - n^n)];
%Y Cf. A309289 (binomial(2k,k) - k^2 is prime), A309290 (binomial(k^2,k) - k^2 is squarefree).
%K nonn,hard,more
%O 1,1
%A _M. F. Hasler_ (at the suggestion of _Juri-Stepan Gerasimov_ and others), Jul 31 2019
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