A055773 a(n) = Product_{p in P_n} where P_n = {p prime, n/2 < p <= n }.

1, 1, 2, 6, 3, 15, 5, 35, 35, 35, 7, 77, 77, 1001, 143, 143, 143, 2431, 2431, 46189, 46189, 46189, 4199, 96577, 96577, 96577, 7429, 7429, 7429, 215441, 215441, 6678671, 6678671, 6678671, 392863, 392863, 392863, 14535931, 765049, 765049, 765049

Offset: 0

Author

Labos Elemer, Jul 12 2000

Keywords

• nonn

Comments

Old name: Product of primes p for which p divides n! but p^2 does not (i.e. ord_p(n!)=1). - Dion Gijswijt (gijswijt(AT)science.uva.nl), Jan 07 2007

Squarefree part of n! divided by gcd(Q,F), where Q is the largest square divisor and F is the squarefree part of n!. - Labos Elemer, Jul 12 2000

a(1) = 1, a(n) = n*a(n-1) if n is a prime else a(n) = least integer multiple of a(n-1)/n. - Amarnath Murthy, Apr 29 2004

Let P(i) denote the primorial number A034386(i). Then a(n) = P(n)/P(floor(n/2)). - Peter Luschny, Mar 05 2011

Letting H(n) = 1 + 1/2 + ... + 1/n denote the n-th harmonic number, it is known that a(n) is equal to the denominator (in lowest terms) of H(n)^2*n! for n >= 6 (see below example). - John M. Campbell, Mar 27 2016

For all n satisfying 6 <= n < 897, a(n) = A130087(n). - John M. Campbell, Mar 27 2016

It is also known that a(n) is equal to lcm^2(1, 2, ..., n)/gcd(lcm^2(1, 2, ..., n), n!). - John M. Campbell, Apr 04 2016

Examples

			n = 13, P_n = {7, 11, 13}, a(13) = 7*11*13 = 1001.
Letting n = 14, the denominator (in lowest terms) of H(n)^2*n! = 131803989435744/143 is a(14)=143. - _John M. Campbell_, Mar 27 2016
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200 (corrected by Michel Marcus, Jan 19 2019)
• J. M. Campbell et al., A problem involving the product prod_{k=1..n} k^mu(k), where mu denotes the Möbius function, Mathematics Stack Exchange (2016).
• F. J. van de Bult, D. C. Gijswijt, J. P. Linderman, N. J. A. Sloane and Allan Wilks, A Slow-Growing Sequence Defined by an Unusual Recurrence, J. Integer Sequences, Vol. 10 (2007), #07.1.2.
• Index entries for sequences related to Gijswijt's sequence

Crossrefs

Cf. A000188, A008833, A007913, A055229, A055231 (for n), A055071, A055204, A055230, A094299, A094302, A193477, A130087.

Programs

Maple

a := n -> mul(k,k=select(isprime,[$iquo(n,2)+1..n])); # Peter Luschny, Jun 20 2009
A055773 := n -> numer(n!/iquo(n,2)!^4); # Peter Luschny, Jul 30 2011

Mathematica

Table[Numerator[n!/Floor[n/2]!^4], {n, 0, 40}] (* Michael De Vlieger, Mar 27 2016 *)

PARI

q=1;for(n=2,41,print1(q,",");q=if(isprime(n),q*n,q/gcd(q,n))) \\ Klaus Brockhaus, May 02 2004

PARI

a(n) = k=1;forprime(p=nextprime(n\2+1),precprime(n),k=k*p);k \\ Klaus Brockhaus, May 02 2004

PARI

a(n) = prod(i=primepi(n/2)+1,primepi(n),prime(i)) \\ John M. Campbell, Mar 27 2016

Python

from math import prod
from sympy import primerange
def A055773(n): return prod(primerange((n>>1)+1,n+1)) # Chai Wah Wu, Apr 13 2024

Formula

a(n) = numerator(A056040(n)^2/n!).

a(n) = numerator(A056040(n)/floor(n/2)!^2).

a(n) = numerator(n!/floor(n/2)!^4). - Peter Luschny, Jul 30 2011

a(n) = product of primes p such that n/2 < p <= n. - Klaus Brockhaus, May 02 2004

a(n) = A055204(n)/A055230(n) = A055231(n!) = A007913(n!)/A055229(n!).

a(n) = Product_{i=pi(n/2)+1..pi(n)} prime(i), where pi denotes the prime counting function and prime(i) denotes the i-th prime number. - John M. Campbell, Mar 27 2016

Extensions

Entry revised by N. J. A. Sloane, Jan 07 2007

Simpler definition based on a comment of Klaus Brockhaus, set offset to 0 and prepended 1 to data. - Peter Luschny, Mar 09 2013