A189914 a(n) is 2^phi(n) times the least common multiple of the proper divisors of n.

1, 2, 2, 4, 8, 16, 24, 64, 64, 192, 160, 1024, 192, 4096, 896, 3840, 2048, 65536, 1152, 262144, 5120, 86016, 22528, 4194304, 6144, 5242880, 106496, 2359296, 114688, 268435456, 7680, 1073741824, 1048576, 34603008, 2228224, 587202560, 147456, 68719476736, 9961472

Offset: 0

Author

Peter Luschny, Jun 22 2011

Keywords

• nonn

Comments

The sequence relates arithmetic properties of roots of unity in the complex plane with number theoretic properties of integers. This connection often appears as intriguing identities showing products of specific values of the sine function or the gamma function reducing to simple values (see for instance the first formula below).

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..3322
• 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], Sep 2009.
• Albert Nijenhuis, Short Gamma Products with Simple Values, The American Mathematical Monthly, Vol. 117, No. 8, Oct 2010, pp. 733-737.
• J. Sándor and L. Tóth, A remark on the gamma function, Elemente der Mathematik, 44 (1989), pp. 73-76, Birkhäuser.

Programs

Maple

A189914 := n -> 2^numtheory[phi](n)*ilcm(op(numtheory[divisors](n) minus {1,n})): seq(A189914(n), n=0..35);

Mathematica

a[n_] := 2^EulerPhi[n] * LCM @@ Most[Divisors[n]]; a[0] = 1; a[1] = 2; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Jan 22 2014 *)

PARI

a(n)=if(n,my(p=n); if(isprime(n)||(ispower(n, , &p)&&isprime(p)), n/p, n)<Charles R Greathouse IV, Jun 24 2011

Formula

Let R(n) = {k | gcd(n,k) = 1, k = 1..floor(n/2)} and b(n) = product_{R(n)} sin(Pi*k/n) then a(n) = n / b(n)^2 for n > 1.

a(n) = A066781(n)*A048671(n).