A263922 -id:A263922 - cp's OEIS frontend

A263931 a(n) = binomial(2n, n) / Product(p prime | n < p <= 2n).

1, 1, 2, 4, 2, 36, 12, 24, 90, 20, 4, 168, 28, 1400, 5400, 720, 90, 5940, 23100, 46200, 180180, 17160, 1560, 140400, 11700, 45864, 179928, 13328, 52360, 5969040, 397936, 795872, 3133746, 12345060, 726180, 2863224, 159068, 318136, 1255800, 4958800, 247940

Offset: 0

Peter Luschny, Oct 31 2015

nonn

The highest exponent in the prime factorization of a(n) is A263922(n), n>=2.
a(n) is even for n>=2.
By the Erdős squarefree conjecture, proved in 1996, no a(n) with n >= 5 is squarefree. - Robert FERREOL, Sep 06 2022
Conjecture: n=5 is the only n for which a(n)^2 > binomial(2*n,n). - Richard Peterson, Dec 18 2024

David A. Corneth, Table of n, a(n) for n = 0..5806 (terms < 10^1000)
Eric Weisstein's World of Mathematics, Erdős Squarefree Conjecture.

Cf. A000984, A261130, A263922, A356637.

a := n -> binomial(2*n,n)/convert(select(isprime, {$n+1..2*n}),`*`):
seq(a(n), n=0..40);

a(n) = { my(res = 1); forprime(p = 2, n, res*= p^(val(2*n, p) - 2*val(n, p))); forprime(p = n + 1, 2*n, res*= p^(val(2*n, p) - 2*val(n, p) - 1)); res }
val(n, p) = my(r=0); while(n, r+=n\=p);r \\ David A. Corneth, Apr 03 2021

from math import comb
from sympy import primorial
def A263931(n): return comb(m:=n<<1,n)*primorial(n,nth=False)//primorial(m,nth=False) if n else 1 # Chai Wah Wu, Sep 07 2022

a(n) = A000984(n)/A261130(n).

A263924 Numbers n such that there is a prime p > 3 and an exponent e such that the central binomial coefficient binomial(2n, n) is divisible by p^e but not by either 2^e or 3^e.

Original entry on oeis.org

64, 256, 272, 324, 513, 514, 516, 544, 1026, 1028, 1032, 1064, 1088, 1089, 1216, 1544, 1552, 1568, 1569, 2052, 2056, 2064, 2188, 2192, 2193, 2194, 2208, 2224, 2244, 2248, 2304, 2313, 2314

Offset: 1

Charles R Greathouse IV, Oct 29 2015

nonn

How quickly does this sequence grow asymptotically?

A263922(a(n)) > max{A000120(a(n)),A000989(an(n))}. - Reinhard Zumkeller, Nov 19 2015

			64 is a member because binomial(128,64) = 2 * 3 * 5^3 * ..., where the exponent 3 of 5 is greater than the exponents 1 and 1 of 2 and 3, respectively.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A263922, A000984, A000120, A000989.

import Math.NumberTheory.Primes.Factorisation (factorise)
a263924 n = a263924_list !! (n-1)
a263924_list = filter f [2..] where
   f x = not (null pe23s) && any ((> e23) . snd) pes' where
         e23 = maximum (map snd pe23s)
         (pe23s, pes') = span ((<= 3) . fst) $ factorise $ a000984 x
-- Reinhard Zumkeller, Nov 01 2015

f(n,p)=my(d=Vecrev(digits(n,p)),c);sum(i=1,#d,c=(2*d[i]+c>=p))
is(n)=my(r=max(hammingweight(n),f(n,3))); forprime(p=5,sqrtnint(n,r+1), if(f(n,p)>r, return(p))); 0

a(n) >> n^1.014. (This is surely not optimal.) - Charles R Greathouse IV, Jan 18 2016

A261129 Highest exponent in prime factorization of the swinging factorial (A056040).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 1, 2, 2, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 2, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 2, 2, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 4, 4, 5, 5, 2, 2, 2, 2, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 4, 4, 2, 4, 4, 4, 3, 3, 4, 4, 4

Offset: 2

Peter Luschny, Oct 31 2015

nonn

A263922 is a subsequence.

Amiram Eldar, Table of n, a(n) for n = 2..10000

Cf. A000120, A000523, A056040, A263922.

swing := n -> n!/iquo(n,2)!^2:
max_exp := n -> max(seq(s[2], s=ifactors(n)[2])):
seq(max_exp(swing(n)), n=2..88);

a[n_] := Max[FactorInteger[n!/Quotient[n, 2]!^2][[;; , 2]]]; Array[a, 100, 2] (* Amiram Eldar, Jul 29 2023 *)

swing = lambda n: factorial(n)//factorial(n//2)^2
max_exp = lambda n: max(e for p, e in n.factor())
[max_exp(swing(n)) for n in (2..88)]

a(n) = A051903(A056040(n)) for n>=2.

A000120(floor(n/2)) <= a(n) <= A000523(n), (n>=2).

A361077 a(n) = largest sqrt(2n)-smooth divisor of binomial(2n, n).

Original entry on oeis.org

1, 1, 2, 4, 2, 36, 12, 24, 18, 4, 4, 24, 4, 200, 5400, 720, 90, 540, 300, 600, 180, 120, 120, 10800, 900, 3528, 10584, 784, 280, 1680, 112, 224, 882, 2940, 2940, 504, 28, 56, 4200, 19600, 980, 158760, 7560, 75600, 113400, 5040, 35280, 211680, 44100, 1800, 648, 432, 216, 45360, 1680

Offset: 0

Matthieu Pluntz, Mar 01 2023

nonn

The highest exponent in the prime factorization of a(n) is A263922(n), for n >= 2.
a(n) is even for n >= 2.
Binomial(2*n, n)/a(n) and A263931(n)/a(n) are coprime with a(n) and squarefree. By the Erdős squarefree conjecture, proved in 1996, no a(n) with n >= 5 is squarefree.

			binomial(10, 5) = 2^2*3^2*7. 2,3 <= sqrt(10), 7 > sqrt(10) so a(5) = 2^2*3^2 = 36.

Matthieu Pluntz, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Erdős Squarefree Conjecture.

Cf. A263922, A263931.

a(n) = my(m=sqrtint(2*n), f=factor(binomial(2*n, n), m+1)); for (k=1, #f~, if (f[k,1]>m, f[k,1]=1)); factorback(f); \\ Michel Marcus, Mar 03 2023

library(primes)
n = 2*10^4
pp = generate_primes(max = sqrt(n))
npinb = list()
for(p in pp){
  np = rep(0,n)
  for(t in 1:(log(n)/log(p))) np[p*(1:(n/p))] = np[1:(n/p)] + 1 #multiplicities of prime factor p in the integers
  npinb[[as.character(p)]] = cumsum(np)[2*(1:(n/2))] - 2*cumsum(np[1:(n/2)]) #multiplicities of prime factor p in the central binomial coefficients
}
res = rep(1,n/2)
for(p in pp){
  select = p^2 <= 2*(1:(n/2))
  res[select] = res[select]*p^npinb[[as.character(p)]][select]
}
#offset of res is 1

a(n) = binomial(2*n, n) / Product(p prime | p^2 > 2*n, floor(2*n/p) is odd).

cp's OEIS Frontend

A263931 a(n) = binomial(2*n, n) / Product(p prime | n < p <= 2*n).

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A263924 Numbers n such that there is a prime p > 3 and an exponent e such that the central binomial coefficient binomial(2n, n) is divisible by p^e but not by either 2^e or 3^e.

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A261129 Highest exponent in prime factorization of the swinging factorial (A056040).

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A361077 a(n) = largest sqrt(2*n)-smooth divisor of binomial(2*n, n).

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A263931 a(n) = binomial(2n, n) / Product(p prime | n < p <= 2n).

A361077 a(n) = largest sqrt(2n)-smooth divisor of binomial(2n, n).