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

2, 5, 7, 7, 11, 13, 13, 17, 19, 19, 23, 23, 23, 29, 31, 31, 31, 37, 37, 41, 43, 43, 47, 47, 47, 53, 53, 53, 59, 61, 61, 61, 67, 67, 71, 73, 73, 73, 79, 79, 83, 83, 83, 89, 89, 89, 89, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 113, 113, 113, 113, 113, 127, 127, 131, 131

Offset

2,1

Comments

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

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

For n>2, a(n) is the largest prime less than 2*n. - Gennady Eremin, Mar 02 2021

Links

Gennady Eremin, Table of n, a(n) for n = 2..800

Gennady Eremin, Factoring Catalan numbers, arXiv:1908.03752 [math.NT], 2019.

Formula

a(n) = A060308(n) = A060265(n) for n>2.

a(n) = A006530(A000108(n)). - Michel Marcus, Nov 14 2015

G.f.: A(x) - x^2, where A(x) is the g.f. of A060265. - Gennady Eremin, Mar 02 2021

Example

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 + ...

Mathematica

Table[Max[FactorInteger[(2n)!/n!/(n+1)! ]], {n, 2, 100}]
FactorInteger[CatalanNumber[#]][[-1, 1]]&/@Range[2, 70] (* Harvey P. Dale, May 02 2017 *)

Prog

(PARI) a(n) = vecmax(factor(binomial(2*n, n)/(n+1))[, 1]); \\ Michel Marcus, Nov 14 2015
(PARI) a(n)=if(n>2, precprime(2*n), 2) \\ Charles R Greathouse IV, Nov 17 2015
(Python)
from gmpy2 import is_prime
A120303 = [2]
for n in range(3, 801):
    for k in range(2*n-1, n, -2):
        if is_prime(k, n):
            A120303.append(k)
            break
for n in range(len(A120303)):
    print(n+2, A120303[n])  # Gennady Eremin, Mar 17 2021

Crossrefs

Cf. A000108, A006530, A020482, A028334, A060265, A060308, A152765.

Sequence in context: A145378 A069887 A254340 * A093413 A004099 A084959

Adjacent sequences: A120300 A120301 A120302 * A120304 A120305 A120306

Keyword

nonn

Author

Alexander Adamchuk, Jul 13 2006

Status

approved