A048671 a(n) is the least common multiple of the proper divisors of n.

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

Offset: 1

Author

Labos Elemer

Keywords

• nonn
• easy

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).

Examples

			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).
		

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.
• Index entries for sequences related to lcm's

Crossrefs

Cf. A000142, A002944, A003418, A014963, A025527.

Cf. A182936 gives the dual (greatest common divisor).

Programs

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 *)

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

Formula

a(n) = A025527(n)/A025527(n-1).

a(n) = (n*A003418(n-1))/A003418(n).

a(n) = A003418(n-1)/A002944(n). [corrected by Michel Marcus, May 18 2020]

From Henry Bottomley, May 19 2000: (Start)

a(n) = n/A014963(n) = lcm(A052126(n), A032742(n)).

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)

Extensions

New definition based on a comment of David Wasserman by Peter Luschny, Mar 23 2011