A309289 Numbers k such that binomial(2k,k) - k^2 is prime.

2, 3, 5, 9, 13, 27, 47, 59, 111, 547, 923, 1301, 2263, 5189, 45403, 66901

Offset: 1

Author

M. F. Hasler (at the suggestion of Juri-Stepan Gerasimov and others), Jul 21 2019

Keywords

• nonn
• hard
• more

Comments

If a term k > 2 is divisible by a prime p, then all digits in the base-p expansion of k are less than p/2. In particular, 2 is the only even term. - Robert Israel, Aug 02 2019

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

Programs

Maple

filter:= t -> isprime(binomial(2*t,t)-t^2):
select(filter, [2,seq(i,i=3..6000,2)]); # Robert Israel, Aug 02 2019

PARI

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

Extensions

a(15)-a(16) from Michael S. Branicky, Aug 24 2024