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A048671
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Least common multiple of the proper divisors of n.
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12
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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
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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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
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FORMULA
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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
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EXAMPLE
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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].
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MAPLE
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A048671 := n -> ilcm(op(numtheory[divisors](n) minus {1, n}));
seq(A048671(i), i=1..79); # Peter Luschny, Mar 21 2011
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MATHEMATICA
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{1}~Join~Table[LCM @@ Most@ Divisors@ n, {n, 2, 79}] (* Michael De Vlieger, May 01 2016 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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Labos Elemer
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EXTENSIONS
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New definition based on a comment of David Wasserman by Peter Luschny, Mar 23 2011
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STATUS
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approved
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