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

Author

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

Keywords

• nonn
• hard
• more

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

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

Crossrefs

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

Programs

Magma

[0] cat  [n: n in [2..45] | IsSquarefree(Binomial(n^2, n) - n^n)];

Mathematica

Select[Range[0, 50], SquareFreeQ[Binomial[#^2, #] - #^#] &]

PARI

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

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