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

1,1

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

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.

Formula

Erdős proved that a(n) <= 2n.

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A050748 A348638 A079051 * A066935 A042943 A306466

Adjacent sequences: A226044 A226045 A226046 * A226048 A226049 A226050

Keyword

nonn,nice

Author

Charles R Greathouse IV, May 24 2013

Status

approved