OFFSET
0,3
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
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.
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) = Product_{i=pi(n/2)+1..pi(n)} p(i), where pi denotes the prime counting function and p(i) denotes the i-th prime number. - John M. Campbell, Mar 27 2016
EXAMPLE
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
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Labos Elemer, Jul 12 2000
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
STATUS
approved