A263931 -id:A263931 - cp's OEIS frontend

A356637 a(n) = A000265(A263931(n)).

1, 1, 1, 1, 1, 9, 3, 3, 45, 5, 1, 21, 7, 175, 675, 45, 45, 1485, 5775, 5775, 45045, 2145, 195, 8775, 2925, 5733, 22491, 833, 6545, 373065, 24871, 24871, 1566873, 3086265, 181545, 357903, 39767, 39767, 156975, 309925, 61985, 5020785, 239085, 20322225, 160730325

Offset: 0

Peter Luschny, Sep 07 2022

nonn

Let n >= 5. If a(n) is squarefree, then 2 divides binomial(2*n, n) more than once and is the only prime that does so. There is only a finite number of such cases (see A059097).
An efficient algorithm for the calculation is available, which is based on prime factorization. See the SageMath implementation. The main application is the efficient calculation of the central binomial coefficient, which is the product of this sequence, the Glaisher/Gould sequence, and the upper primorial function (see the formula section).
Since the central binomial coefficient is a bisection of the swinging factorial A056040, and the swinging factorial, in turn, is the building block for an efficient algorithm for the computation of the factorial function, the terms of this sequence occur as factors in all these computations. See the links for details.

			Let n = 22 and consider the prime factorization of m = binomial(2*n, n):
2^3 * [3 * 5 * 13] * 23 * 29 * 31 * 37 * 41 * 43. Then a(22) = 3 * 5 * 13. This is what is left after the 'prime tail' A261130(n) and the 'prime head' A006519(m) = A001316(n) have been cut off.

Peter Luschny, Swing, divide and conquer the factorial, (excerpt).
Peter Luschny, Fast Factorial Functions, a code repository.
Eric Weisstein's World of Mathematics, Erdős Squarefree Conjecture.

Cf. A000265, A263931, A261130, A006519, A000984, A001316, A059097, A056040.

A263931 := n -> binomial(2*n, n) / convert(select(isprime, {$n+1..2*n}), `*`):
A000265 := n -> n / 2^padic[ordp](n, 2):
seq(A000265(A263931(n)), n = 0..45);

def A356637(n: int) -> int:
    m = 2 * n
    if m < 5: return 1
    sqrtm = isqrt(m) + 1
    R = prime_range(sqrtm, m // 3 + 1)
    factors = [x for x in R if is_odd(m // x)]
    for prime in prime_range(3, sqrtm):
        p: int = 1
        q: int = m
        while True:
            q //= prime
            if q == 0:
                break
            if q & 1 == 1:
                p *= prime
        if p > 1:
            factors.append(p)
    return product(factors)
print([A356637(n) for n in range(45)])

A000984(n) = a(n) * A001316(n) * A261130(n) for n >= 2.

A261130 a(n) = Product(p prime | n < p <= 2*n).

Original entry on oeis.org

1, 2, 3, 5, 35, 7, 77, 143, 143, 2431, 46189, 4199, 96577, 7429, 7429, 215441, 6678671, 392863, 392863, 765049, 765049, 31367009, 1348781387, 58642669, 2756205443, 2756205443, 2756205443, 146078888479, 146078888479, 5037203051, 297194980009, 584803025179

Offset: 0

Peter Luschny, Oct 31 2015

Essentially the same as A068111. - R. J. Mathar, Nov 23 2015

a(n) is a divisor of binomial(2*n, n); the quotient binomial(2*n, n) / a(n) is A263931(n). - Robert FERREOL, Sep 03 2022

			a(0) = 1 because the empty product is 1 by convention.
a(4) = 35 because {p prime | 4 < p <= 8} = {5, 7}.

Cf. A000984 (binomial(2*n,n)), A034386, A263931, A356637.

a := n -> convert(select(isprime, {$n+1..2*n}),`*`):
print(seq(a(n), n=0..31));

Join[{1},Table[Times@@Prime[Range[PrimePi[n]+1,PrimePi[2n]]],{n,40}]] (* Harvey P. Dale, May 09 2017 *)

A261130(n,P=1)={forprime(p=n+1,2*n,P*=p);P} \\ M. F. Hasler, Nov 25 2015

from sympy import primorial
def A261130(n): return primorial(n<<1,nth=False)//primorial(n,nth=False) if n else 1 # Chai Wah Wu, Sep 07 2022

A135322 a(n) = gcd(n!, binomial(2n,n)).

Original entry on oeis.org

1, 1, 2, 2, 2, 12, 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

Offset: 0

Leroy Quet, Dec 06 2007

nonn

			a(5) = 12 as gcd(5!, binomial(2*5, 5)) = gcd(120, 252) = 12. - _David A. Corneth_, Apr 03 2021

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A000984, A263931.

Table[GCD[n!, Binomial[2n, n]], {n, 0, 60}] (* Stefan Steinerberger, Dec 07 2007 *)

valp(n,p)=my(s); while(n\=p, s+=n); s
a(n)=my(s=1,t); forprime(p=2,n, t=valp(n,p); t=min(t,valp(2*n,p)-2*t); if(t, s*=p^t)); s \\ Charles R Greathouse IV, Oct 09 2016

More terms from Stefan Steinerberger, Dec 07 2007

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

A356637 a(n) = A000265(A263931(n)).

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A261130 a(n) = Product(p prime | n < p <= 2*n).

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A135322 a(n) = gcd(n!, binomial(2n,n)).

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Mathematica

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

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Formula

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