A226083 Smallest element of the set of largest prime powers p^k dividing C(2*n,n), where p is any prime factor of C(2*n,n).

2, 2, 4, 2, 4, 3, 3, 2, 4, 4, 3, 4, 7, 8, 5, 2, 4, 3, 3, 4, 3, 3, 13, 4, 8, 8, 16, 5, 3, 7, 7, 2, 3, 3, 7, 4, 7, 3, 11, 4, 5, 5, 7, 7, 5, 5, 5, 4, 8, 8, 11, 8, 5, 3, 3, 8, 3, 3, 5, 7, 7, 7, 3, 2, 4, 3, 3, 4, 7, 8, 11, 4, 8, 8, 5, 5, 5, 7, 7, 4, 5, 5, 3, 7, 5, 5, 3, 3, 9, 11, 7, 3, 7, 7, 13, 4, 8, 8, 3, 3

Offset: 1

Author

Alois P. Heinz, May 25 2013

Keywords

• nonn
• look

Examples

			a(89) = 9: C(2*89,89) = 2^4 * 3^2 * 5^3 * 7^2 * 11^1 * ... * 173^1, the smallest prime power is 3^2 = 9.  3^2 is the largest prime power for prime 3 dividing C(2*89,89).
a(9993) = 59: 59^1 is the largest power of 59 dividing C(2*9993,9993), it is smaller than the largest powers of all other prime factors.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A000961, A000984, A007318, A226047 (row maxima of A226078).

Programs

Maple

a:= proc(n) local h, i, m, p;
      p:=1; m:=infinity;
      while p < m do p:= nextprime(p); i:= 0;
         h:= 2*n; while h>0 do h:=iquo(h, p); i:=i+h od;
         h:= n;   while h>0 do h:=iquo(h, p); i:=i-2*h od;
         if i>0 then m:= min(m, p^i) fi
      od; m
    end:
seq(a(n), n=1..100);

Formula

a(n) = min_{p prime, p|C(2n,n)} max_{k, p^k|C(2n,n)} p^k.

a(n) = min_{k=0..A067434(n)-1} A226078(n,k).