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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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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 E.]
a(n) is the lcm of the proper divisors of n. - David Wasserman (wasserma(AT)spawar.navy.mil), Nov 30 2004
a(n) = (n^2)/A140580. - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 17 2008
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LINKS
| Peter Luschny and Stefan Wehmeier, The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences
Eric Weisstein's World of Mathematics, Sylvester Cyclotomic Number
Index entries for sequences related to lcm's
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FORMULA
| 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 (se16(AT)btinternet.com), May 19 2000
a(n) = n*Product_{ d divides n } d^mu(d). Product_{ d divides n } a(d) = A007956(n). - Vladeta Jovovic (vladeta(AT)eunet.rs), 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 (pbarry(AT)wit.ie), 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
| 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
| 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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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)
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
| Cf. A025527, A003418, A002944, A000142, A014963, A140580.
Cf. A182936 gives the dual (greatest common divisor).
Sequence in context: A193807 A154744 A189733 * A205959 A088123 A050932
Adjacent sequences: A048668 A048669 A048670 * A048672 A048673 A048674
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KEYWORD
| nonn,easy,changed
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AUTHOR
| Labos E. (labos(AT)ana.sote.hu)
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EXTENSIONS
| New definition based on a comment of David Wasserman. - Peter Luschny, Mar 23 2011.
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