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

0,2

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.

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).

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 *)

Prog

(PARI) a(n)=if(n, my(p=n); if(isprime(n)||(ispower(n, , &p)&&isprime(p)), n/p, n)<<eulerphi(n), 1) \\ Charles R Greathouse IV, Jun 24 2011

Crossrefs

Sequence in context: A005864 A112433 A171648 * A286496 A318187 A217931

Adjacent sequences: A189911 A189912 A189913 * A189915 A189916 A189917

Keyword

nonn

Author

Peter Luschny, Jun 22 2011

Status

approved