|
|
A055773
|
|
a(n) = Product_{p in P_n} where P_n = {p prime, n/2 < p <= n }.
|
|
20
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
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
|
|
|
FORMULA
|
a(n) = numerator(A056040(n)/floor(n/2)!^2).
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
|
|
MATHEMATICA
|
|
|
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
|
|
CROSSREFS
|
Cf. A000188, A008833, A007913, A055229, A055231 (for n), A055071, A055204, A055230, A094299, A094302, A193477, A130087.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|