A309289 - OEIS

%I #23 Aug 24 2024 09:00:04

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

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

%C 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

%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.

%p filter:= t -> isprime(binomial(2*t,t)-t^2):

%p select(filter, [2,seq(i,i=3..6000,2)]); # _Robert Israel_, Aug 02 2019

%o (PARI) is(n)=ispseudoprime(binomial(2*n,n)-n^2)

%o for(n=0,oo, is(n) && print1(n,", "))

%Y Cf. 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 21 2019

%E a(15)-a(16) from _Michael S. Branicky_, Aug 24 2024