|
|
A048671
|
|
a(n) is the least common multiple of the proper divisors of n.
|
|
16
|
|
|
1, 1, 1, 2, 1, 6, 1, 4, 3, 10, 1, 12, 1, 14, 15, 8, 1, 18, 1, 20, 21, 22, 1, 24, 5, 26, 9, 28, 1, 30, 1, 16, 33, 34, 35, 36, 1, 38, 39, 40, 1, 42, 1, 44, 45, 46, 1, 48, 7, 50, 51, 52, 1, 54, 55, 56, 57, 58, 1, 60, 1, 62, 63, 32, 65, 66, 1, 68, 69, 70, 1, 72, 1, 74, 75, 76, 77, 78, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
FORMULA
|
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)
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
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}));
|
|
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
(Sage)
if n < 2 : return 1
else : D = divisors(n); D.pop()
return lcm(D)
|
|
CROSSREFS
|
Cf. A182936 gives the dual (greatest common divisor).
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|