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A048671 Least common multiple of the proper divisors of n. 12
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.

a(n) = q(n)/q(n-1), where q(n) = n!/A003418(n). [Definition as given by Labos Elemer]

a(n) is the lcm of the proper divisors of n. - David Wasserman, Nov 30 2004

a(n) = (n^2)/A140580. - Gary W. Adamson, May 17 2008

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

FORMULA

a(n) = A025527(n)/A025527(n-1) or (n*LCM(n-1))/LCM(n) where LCM(n) is least common multiple of first n natural numbers: LCM(n) = A003418(n).

Also a(n) = A003418(n)/A002944(n)=LCM[1, .., n]/LCM[.., C[n, j], ..].

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) - Henry Bottomley, May 19 2000

a(n) = n*Product_{ d divides n } d^mu(d). Product_{ d divides n } a(d) = A007956(n). - Vladeta Jovovic, Jul 04 2002

a(n) = product{k=1..n-1, if(gcd(n, k)>1, 1-exp(2*pi*I*k/n), 1)}, I=sqrt(-1); - Paul Barry, Apr 15 2005

a(n) = product{0<k<n} (if(gcd(k,n)>1, 2*Pi/Gamma(k/n)^2, 1);  a(n) = product{0<k<n} (if(gcd(k,n)>1, 2*sin(Pi*k/n), 1). - Peter Luschny, Jun 09 2011

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

Cf. A025527, A003418, A002944, A000142, A014963, A140580.

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

Sequence in context: A189733 A277791 A243146 * A205959 A088123 A050932

Adjacent sequences:  A048668 A048669 A048670 * A048672 A048673 A048674

KEYWORD

nonn,easy

AUTHOR

Labos Elemer

EXTENSIONS

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

STATUS

approved

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Last modified March 24 00:37 EDT 2017. Contains 283984 sequences.