A226047 Largest prime power dividing binomial(2n, n).

2, 3, 5, 7, 9, 11, 13, 13, 17, 19, 19, 23, 25, 27, 29, 31, 31, 31, 37, 37, 41, 43, 43, 47, 49, 49, 53, 53, 53, 59, 61, 61, 61, 67, 67, 71, 73, 73, 73, 79, 81, 83, 83, 83, 89, 89, 89, 89, 97, 97, 101, 103, 103, 107, 109, 109, 113, 113, 113, 113

Offset: 1

Author

Charles R Greathouse IV, May 24 2013

Keywords

• nonn
• nice

Comments

The first indices n for which a(n) < a(n-1) are 123, 315, 366, 671, 1095, 1098, 1204, 1565, 6095, 7326, 9843, 39065, 58828, 88575, 88578, 195315, 195320, 265722, 265725 and 709937. - Giovanni Resta, May 24 2013

a(n) = maximum of n-th row in A226078. - Reinhard Zumkeller, May 25 2013

Examples

			Binomial(10, 5) = 2^2 * 3^2 * 7 and so a(5) = max({2^2, 3^2, 7}) = 3^2.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• P. Erdős, Beweis eines Satzes von Tschebyschef (in German), Acta Litt. Sci. Szeged 5 (1932), pp. 194-198.

Crossrefs

Cf. A000961, A000984, A060308, A067434, A226083.

Programs

Haskell

a226047 = maximum . a226078_row  -- Reinhard Zumkeller, May 25 2013

Maple

f:= n-> add(i[2]*x^i[1], i=ifactors(n)[2]):
a:= proc(n) local p;
      p:= add(f(n+i) -f(i), i=1..n);
      max(seq(i^coeff(p, x, i), i=1..degree(p)))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, May 24 2013

Mathematica

cnt[n_, p_] := (n - Total@IntegerDigits[n, p])/(p-1); a[n_] := Block[{k = 2*n, p, e}, While[! PrimePowerQ[k] || ({p, e} = FactorInteger[k][[1]]; cnt[2*n , p] - 2 cnt[n, p] != e), k--]; k]; Array[a, 60] (* Giovanni Resta, May 24 2013 *)
Table[Max[Select[Divisors[Binomial[2 n,n]],PrimePowerQ]],{n,60}] (* Harvey P. Dale, Feb 26 2024 *)

PARI

ord(n,p)=my(s);while(n\=p,s+=n);s
a(n)=my(p=precprime(2*n));forstep(k=2*n,p+1,-1, my(q,e=isprimepower(k, &q)); if(e && e == ord(2*n,q)-2*ord(n,q), return(k)));p /* requires PARI v.2.5 or later */

PARI

A226047(n)={for(k=2,#n=factor(binomial(2*n,n))~,factorback(n[,k-1]~)>factorback(n[,k]~) && n[,k]=n[,k-1]);factorback(n[,#n]~)} \\ highly unoptimized, not suitable for n>>10^4. - M. F. Hasler, May 24 2013

Formula

Erdős proved that a(n) <= 2n.