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A226047 Largest prime power dividing binomial(2n, n). 4
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 (list; graph; refs; listen; history; text; internal format)
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 *)

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: A061979 A050748 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

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Last modified July 4 14:29 EDT 2020. Contains 335448 sequences. (Running on oeis4.)