cp's OEIS Frontend

A120303 Largest prime factor of Catalan number A000108(n).

%I A120303 #43 Apr 16 2021 09:51:40
%S A120303 2,5,7,7,11,13,13,17,19,19,23,23,23,29,31,31,31,37,37,41,43,43,47,47,
%T A120303 47,53,53,53,59,61,61,61,67,67,71,73,73,73,79,79,83,83,83,89,89,89,89,
%U A120303 97,97,101,103,103,107,109,109,113,113,113,113,113,113,113,127,127,131,131
%N A120303 Largest prime factor of Catalan number A000108(n).
%C A120303 All prime numbers (except 3) are present in this sequence in their natural order with repetition. The number of repetitions is equal to A028334(n): differences between consecutive primes, divided by 2. - _Alexander Adamchuk_, Jul 30 2006
%C A120303 For p>3 a((p+1)/2) = p and all a(n) = p for n >= (p+1)/2 until the first occurrence of the next prime q = NextPrime(p) at a((q+1)/2) = q. - _Alexander Adamchuk_, Dec 27 2013
%C A120303 For n>2, a(n) is the largest prime less than 2*n. - _Gennady Eremin_, Mar 02 2021
%H A120303 Gennady Eremin, <a href="/A120303/b120303.txt">Table of n, a(n) for n = 2..800</a>
%H A120303 Gennady Eremin, <a href="https://arxiv.org/abs/1908.03752">Factoring Catalan numbers</a>, arXiv:1908.03752 [math.NT], 2019.
%F A120303 a(n) = A060308(n) = A060265(n) for n>2.
%F A120303 a(n) = A006530(A000108(n)). - _Michel Marcus_, Nov 14 2015
%F A120303 G.f.: A(x) - x^2, where A(x) is the g.f. of A060265. - _Gennady Eremin_, Mar 02 2021
%e A120303 G.f. = 2*x^2 + 5*x^3 + 7*x^4 + 7*x^5 + 11*x^6 + 13*x^7 + 13*x^8 + 17*x^9 + ...
%t A120303 Table[Max[FactorInteger[(2n)!/n!/(n+1)! ]],{n,2,100}]
%t A120303 FactorInteger[CatalanNumber[#]][[-1,1]]&/@Range[2,70] (* _Harvey P. Dale_, May 02 2017 *)
%o A120303 (PARI) a(n) = vecmax(factor(binomial(2*n, n)/(n+1))[,1]); \\ _Michel Marcus_, Nov 14 2015
%o A120303 (PARI) a(n)=if(n>2,precprime(2*n),2) \\ _Charles R Greathouse IV_, Nov 17 2015
%o A120303 (Python)
%o A120303 from gmpy2 import is_prime
%o A120303 A120303 = [2]
%o A120303 for n in range(3, 801):
%o A120303     for k in range(2*n-1, n, -2):
%o A120303         if is_prime(k, n):
%o A120303             A120303.append(k)
%o A120303             break
%o A120303 for n in range(len(A120303)):
%o A120303     print(n+2, A120303[n])  # _Gennady Eremin_, Mar 17 2021
%Y A120303 Cf. A000108, A006530, A020482, A028334, A060265, A060308, A152765.
%K A120303 nonn
%O A120303 2,1
%A A120303 _Alexander Adamchuk_, Jul 13 2006