OFFSET
1,4
COMMENTS
A proper divisor d of n is a divisor of n such that 1 <= d < n.
Previous name was: a(n) = q(n)/q(n-1), where q(n) = n!/A003418(n).
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..10000
Peter Luschny and Stefan Wehmeier, The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], 2009.
Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number.
FORMULA
From Henry Bottomley, May 19 2000: (Start)
a(n) = n if n not a prime power, a(n) = n/p if n = p^m (i.e., a(n) = 1 if n = p). (End)
From Vladeta Jovovic, Jul 04 2002: (Start)
a(n) = n*Product_{d | n} d^mu(d).
Product_{d | n} a(d) = A007956(n). (End)
a(n) = Product_{k=1..n-1} if(gcd(n, k) > 1, 1 - exp(2*pi*i*k/n), 1), where i = sqrt(-1). - Paul Barry, Apr 15 2005
From Peter Luschny, Jun 09 2011: (Start)
a(n) = Product_{k=1..n-1} if(gcd(k,n) > 1, 2*Pi/Gamma(k/n)^2, 1).
a(n) = Product_{k=1..n-1} if(gcd(k,n) > 1, 2*sin(Pi*k/n), 1). (End)
EXAMPLE
8!/lcm(8) = 48 = 40320/840 while 7!/lcm(7) = 5040/420 = 12 so a(8) = 48/12 = 4.
a(5) = 1 = lcm(1,2,3,4,5)/lcm(1,5,10,10,5,1).
MAPLE
A048671 := n -> ilcm(op(numtheory[divisors](n) minus {1, n}));
seq(A048671(i), i=1..79); # Peter Luschny, Mar 21 2011
MATHEMATICA
{1}~Join~Table[LCM @@ Most@ Divisors@ n, {n, 2, 79}] (* Michael De Vlieger, May 01 2016 *)
PROG
(PARI) a(n)=my(p=n); if(isprime(n)||(ispower(n, , &p)&&isprime(p)), n/p, n) \\ Charles R Greathouse IV, Jun 24 2011
(PARI) a(n)=my(p); if(isprimepower(n, &p), n/p, n) \\ Charles R Greathouse IV, May 02 2016
(Sage)
def A048671(n) :
if n < 2 : return 1
else : D = divisors(n); D.pop()
return lcm(D)
[A048671(i) for i in (1..79)] # Peter Luschny, Feb 03 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
New definition based on a comment of David Wasserman by Peter Luschny, Mar 23 2011
STATUS
approved