cp's OEIS Frontend

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

%I A048671 #62 Feb 16 2025 08:32:40
%S A048671 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,
%T A048671 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,
%U A048671 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
%N A048671 a(n) is the least common multiple of the proper divisors of n.
%C A048671 A proper divisor d of n is a divisor of n such that 1 <= d < n.
%C A048671 Previous name was: a(n) = q(n)/q(n-1), where q(n) = n!/A003418(n).
%H A048671 Michael De Vlieger, <a href="/A048671/b048671.txt">Table of n, a(n) for n = 1..10000</a>
%H A048671 Peter Luschny and Stefan Wehmeier, <a href="http://arxiv.org/abs/0909.1838">The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences</a>, arXiv:0909.1838 [math.CA], 2009.
%H A048671 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SylvesterCyclotomicNumber.html">Sylvester Cyclotomic Number</a>.
%H A048671 <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>
%F A048671 a(n) = A025527(n)/A025527(n-1).
%F A048671 a(n) = (n*A003418(n-1))/A003418(n).
%F A048671 a(n) = A003418(n-1)/A002944(n). [corrected by _Michel Marcus_, May 18 2020]
%F A048671 From _Henry Bottomley_, May 19 2000: (Start)
%F A048671 a(n) = n/A014963(n) = lcm(A052126(n), A032742(n)).
%F A048671 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)
%F A048671 From _Vladeta Jovovic_, Jul 04 2002: (Start)
%F A048671 a(n) = n*Product_{d | n} d^mu(d).
%F A048671 Product_{d | n} a(d) = A007956(n). (End)
%F A048671 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
%F A048671 From _Peter Luschny_, Jun 09 2011: (Start)
%F A048671 a(n) = Product_{k=1..n-1} if(gcd(k,n) > 1, 2*Pi/Gamma(k/n)^2, 1).
%F A048671 a(n) = Product_{k=1..n-1} if(gcd(k,n) > 1, 2*sin(Pi*k/n), 1). (End)
%e A048671 8!/lcm(8) = 48 = 40320/840 while 7!/lcm(7) = 5040/420 = 12 so a(8) = 48/12 = 4.
%e A048671 a(5) = 1 = lcm(1,2,3,4,5)/lcm(1,5,10,10,5,1).
%p A048671 A048671 := n -> ilcm(op(numtheory[divisors](n) minus {1,n}));
%p A048671 seq(A048671(i), i=1..79); # _Peter Luschny_, Mar 21 2011
%t A048671 {1}~Join~Table[LCM @@ Most@ Divisors@ n, {n, 2, 79}] (* _Michael De Vlieger_, May 01 2016 *)
%o A048671 (PARI) a(n)=my(p=n);if(isprime(n)||(ispower(n,,&p)&&isprime(p)),n/p,n) \\ _Charles R Greathouse IV_, Jun 24 2011
%o A048671 (PARI) a(n)=my(p); if(isprimepower(n,&p), n/p, n) \\ _Charles R Greathouse IV_, May 02 2016
%o A048671 (Sage)
%o A048671 def A048671(n) :
%o A048671     if n < 2 : return 1
%o A048671     else : D = divisors(n); D.pop()
%o A048671     return lcm(D)
%o A048671 [A048671(i) for i in (1..79)] # _Peter Luschny_, Feb 03 2012
%Y A048671 Cf. A000142, A002944, A003418, A014963, A025527.
%Y A048671 Cf. A182936 gives the dual (greatest common divisor).
%K A048671 nonn,easy
%O A048671 1,4
%A A048671 _Labos Elemer_
%E A048671 New definition based on a comment of _David Wasserman_ by _Peter Luschny_, Mar 23 2011