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

2, 3, 5, 7, 13, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 77, 79, 83, 85, 89, 97

Offset

1,1

Comments

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?

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.

97 is a term (see SeqFan list discussion). - Chai Wah Wu, Jul 20 2020

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

Links

Table of n, a(n) for n=1..29.

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, #] - #^#] &]

Prog

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

Crossrefs

Cf. A309289 (binomial(2k,k) - k^2 is prime), A309290 (binomial(k^2,k) - k^2 is squarefree).

Sequence in context: A233008 A233009 A243535 * A050757 A059168 A032758

Adjacent sequences: A308075 A308076 A308077 * A308079 A308080 A308081

Keyword

nonn,hard,more

Author

M. F. Hasler (at the suggestion of Juri-Stepan Gerasimov and others), Jul 31 2019

Extensions

a(26) from Max Alekseyev, Mar 09 2025

a(27) from Tyler Busby, Mar 10 2025

a(28)-a(29) from Tyler Busby, Apr 16 2025

Status

approved