A067434 -id:A067434 - cp's OEIS frontend

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

Alois P. Heinz, May 25 2013

			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.

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

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

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);

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).

A383213 a(n) = number of distinct prime factors of binomial(2n,n+1).

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 4, 4, 5, 5, 6, 6, 6, 7, 6, 7, 9, 8, 10, 9, 10, 10, 10, 9, 10, 10, 11, 11, 12, 13, 12, 12, 14, 14, 14, 14, 14, 14, 16, 14, 16, 15, 16, 17, 16, 17, 18, 17, 18, 18, 18, 18, 20, 18, 20, 19, 19, 20, 20, 21, 21, 21, 21, 21, 23, 22, 24, 23, 23, 23

Offset: 1

Clark Kimberling, Apr 19 2025

nonn

			binomial(6,4)= 3*5, so a(3)=2.

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A000984, A001221, A067434, A383214.

Table[PrimeNu[Binomial[2 n, n + 1]], {n,  200}]

a(n) = omega(binomial(2*n,n+1)); \\ Michel Marcus, Apr 19 2025

from collections import Counter
from sympy import factorint
def A383213(n): return len(sum((Counter(factorint(i)) for i in range(n+2,(n<<1)+1)),start=Counter())-sum((Counter(factorint(i)) for i in range(1,n)),start=Counter())) # Chai Wah Wu, Apr 26 2025

A067437 Number of distinct prime factors in binomial(2*n,n)+1.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 1, 2, 3, 2, 2, 1, 2, 3, 3, 4, 2, 2, 1, 2, 4, 1, 3, 2, 4, 4, 3, 4, 3, 2, 3, 2, 2, 2, 5, 4, 3, 1, 4, 3, 2, 3, 2, 4, 2, 1, 3, 3, 4, 2, 6, 3, 7, 5, 4, 5, 3, 3, 2, 3, 5, 1, 5, 2, 3, 4, 3, 1, 6, 6, 3, 1, 3, 2, 5, 2, 4, 4, 4, 2, 6, 2, 2, 1, 3, 4, 3, 2, 4, 3

Offset: 1

Benoit Cloitre, Feb 23 2002

Amiram Eldar, Table of n, a(n) for n = 1..212

Cf. A001221, A000984, A067434, A244174.

a[n_] := PrimeNu[Binomial[2*n,n] + 1]; Array[a,90] (* Amiram Eldar, Apr 23 2022 *)

a(n) = A001221(A244174(n)). - Amiram Eldar, Apr 23 2022

A071853 Numbers n such that for any x, C(2x,x) never has n distinct prime factors.

Original entry on oeis.org

11, 38, 51, 70, 84, 92, 107, 147, 155, 167, 279, 310, 326, 354, 377, 403, 469, 475, 489, 534, 588, 607, 619, 669, 685, 687, 710, 783, 805, 820, 848, 898, 947, 1010, 1101, 1207, 1332, 1467, 1481, 1520, 1556, 1578, 1738, 1744, 1796, 1802

Offset: 1

Benoit Cloitre, Jun 09 2002

nonn

n such that Card ( x : A067434(x) = n ) = 0

More terms from Naohiro Nomoto, Jun 16 2002

A071854 a(n) = (number of distinct prime factors in C(2n,n)) - pi(n).

Original entry on oeis.org

1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 1, 2, 1, 2, 1, 2, 2, 2, 0, 1, 1, 1, 1, 3, 3, 2, 1, 2, 3, 3, 3, 3, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 4, 3, 3, 3, 4, 4, 5, 5, 4, 4, 4, 3, 4, 3, 3, 4, 4, 5, 5, 4, 5, 5, 5, 4, 4, 4, 5, 5, 5, 4, 4, 5, 6, 5, 5, 5, 6, 5, 4, 5, 6, 5, 6, 5, 6, 6

Offset: 1

Benoit Cloitre, Jun 09 2002

nonn

T. D. Noe, Table of n, a(n) for n=1..1000

for(n=1,200,print1(omega(binomial(2*n,n))-sum(i=1,n,isprime(i)),","))

a(n) = A067434(n) - A000720(n)

A129515 Numbers m such that binomial(2m, m) has the same prime factors as binomial(2k, k) for some k > m.

Original entry on oeis.org

87, 199, 237, 467, 607, 967, 1127, 1319, 1483, 1903, 1943, 2012, 2047, 2287, 2348, 2359, 2464, 2479, 2495, 2507, 2623, 2645, 2719, 3349, 3467, 3514, 3568, 3629, 3633, 3712, 3847, 3919, 4088, 4224, 4287, 4360, 4479, 4927, 4987, 5087, 5167, 5224, 5669

Offset: 1

T. D. Noe, Apr 18 2007

The Erdős paper mentions 87 and 607. The paper conjectures that the sequence is infinite. For the m listed here, k=m+1. Note that we need only examine k such that pi(2*m) = pi(2*k), where pi is the prime counting function.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (corrected and extended from original by T. D. Noe)
Thomas Bloom, Problem 730, Erdős Problems.
P. Erdős, R. L. Graham, I. Z. Russa and E. G. Straus, On the prime factors of C(2n,n), Math. Comp. 29 (1975), 83-92.

Cf. A067434 (number of distinct prime factors in binomial(2n, n)).

s={}; nLst={}; t={}; Do[p=Transpose[FactorInteger[Binomial[2n,n]]][[1]]; If[s!={} && p[[ -1]]!=s[[ -1,-1]], s={}; nLst={}]; pos=Position[s,p,1,1]; If[pos!={}, m=pos[[1,1]]; AppendTo[t,nLst[[m]]], AppendTo[s,p]; AppendTo[nLst,n]], {n,10000}]; t

valp(n,p)=my(s); while(n\=p, s+=n); s
f(n,p)=valp(2*n,p)==2*valp(n,p)
is(n)=for(k=n+1,nextprime(2*n)\2, forprime(p=2,2*n, if(f(n,p)!=f(k,p), next(2))); return(k)); 0 \\ Charles R Greathouse IV, Oct 18 2017

A213609 Smallest number k such that the number of distinct prime divisors of binomial(2k,k) equals n, otherwise 0.

Original entry on oeis.org

1, 2, 4, 6, 8, 11, 15, 16, 18, 20, 0, 28, 29, 33, 38, 42, 45, 48, 53, 54, 60, 64, 66, 67, 75, 77, 80, 86, 91, 92, 100, 102, 104, 109, 111, 110, 127, 0, 128, 133, 140, 144, 151, 154, 153, 160, 165, 170, 171, 178, 0, 189, 190, 192, 198, 202, 209, 210, 220, 225

Offset: 1

Michel Lagneau, Jun 16 2012

nonn

a(A071853(n)) = 0.

			a(3) = 4 because binomial(2*4,4) = 70 with 3 distinct prime divisors {2, 5, 7}.

Olivier Gérard, Table of n, a(n) for n = 1..1000

Cf. A067434, A071853.

with(numtheory): for n from 1 to 100 do:ii:=0: for k from 1 to 500 while(ii=0) do:x:=binomial(2*k,k):y:=factorset(x): n1:=nops(y):if n1=n then ii:=1:printf(`%d, `,k):else fi:od:if ii=0 then printf(`%d, `,0):else fi:od:

A335209 Numbers k such that binomial(2k,k) has more distinct prime factors than binomial(2k,i) for 0 <= i < k.

Original entry on oeis.org

1, 2, 4, 6, 8, 20, 32, 54, 66, 110, 144, 170, 200, 210, 278, 288, 304, 330, 402, 405, 468, 510, 527, 628, 654, 684, 704, 778, 783, 784, 853, 891, 892, 990, 1001, 1035, 1125, 1155, 1232, 1296, 1334, 1384, 1394, 1488, 1495, 1521, 1551, 1575, 1600, 1625, 1645, 1701, 1768, 1875, 1891, 2028, 2072

Offset: 1

Robert Israel, May 26 2020

nonn

Numbers k such that A020733(2*k) = 1.

			a(4)=6 is in the sequence because binomial(12,6) = 924 = 2^2*3*7*11 has 4 distinct prime factors while binomial(12,0) to binomial(12,5) all have at most 3.
7 is not in the sequence because binomial(14,7) = 3432 = 2^3*3*11*13 and binomial(14,6) = 3003 = 3*7*11*13 both have 4 distinct prime factors.

Robert Israel, Table of n, a(n) for n = 1..250

Cf. A001221, A020733, A067434.

filter:=  proc(n) local t, v, i, m;
  m:= 0: t:= 1:
  for i from 1 to n-1 do
   t:= t * ifactor(2*n-i+1)/ifactor(i);
   if type(t,`*`) then v:= nops(t) else v:= 1 fi;
   if v > m then m:= v fi;
  od;
  t:= t*ifactor(n+1)/ifactor(n);
  type(t,`*`) and nops(t) > m
end proc:
filter(1):= true:
select(filter, [$1..2500]); # Robert Israel, May 26 2020

Select[Range@ 1001, Max@ Most@ # < Last@ # &@ PrimeNu@ Binomial[2 #, Range[0, #]] &] (* Michael De Vlieger, May 26 2020 *)

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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).

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A383213 a(n) = number of distinct prime factors of binomial(2n,n+1).

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A067437 Number of distinct prime factors in binomial(2*n,n)+1.

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A071853 Numbers n such that for any x, C(2x,x) never has n distinct prime factors.

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A071854 a(n) = (number of distinct prime factors in C(2n,n)) - pi(n).

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A129515 Numbers m such that binomial(2*m, m) has the same prime factors as binomial(2*k, k) for some k > m.

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A213609 Smallest number k such that the number of distinct prime divisors of binomial(2k,k) equals n, otherwise 0.

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A335209 Numbers k such that binomial(2*k,k) has more distinct prime factors than binomial(2*k,i) for 0 <= i < k.

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

A129515 Numbers m such that binomial(2m, m) has the same prime factors as binomial(2k, k) for some k > m.

A335209 Numbers k such that binomial(2k,k) has more distinct prime factors than binomial(2k,i) for 0 <= i < k.