A064446 a(n) = gcd(n!, n^n, lcm(1, 2, ..., n)), or gcd(n^n, lcm(1, 2, ..., n)).

1, 2, 3, 4, 5, 12, 7, 8, 9, 40, 11, 72, 13, 56, 45, 16, 17, 144, 19, 80, 63, 176, 23, 144, 25, 208, 27, 112, 29, 10800, 31, 32, 297, 544, 175, 864, 37, 608, 351, 800, 41, 6048, 43, 352, 675, 736, 47, 864, 49, 800, 459, 416, 53, 864, 275, 1568, 513, 928, 59, 21600, 61

Offset: 1

Author

Labos Elemer, Oct 02 2001

Keywords

• nonn

Comments

gcd(n^n, lcm(1..n)) must be limited to products of all the distinct prime divisors p of n. We can regard lcm(1..n) as the product of a "regular" factor r produced by primes that also divide n and a coprime factor s produced by primes that are coprime to n. Since the distinct prime divisors p of n are the only distinct prime divisors of n^n, we need only consider r and can ignore s when considering gcd(n^n, lcm(1..n)). Because r is the product of the largest power e_1 of each distinct prime divisor p, and since the power e_2 of the corresponding primes that divide n^n must always be such that e_2 >= e_1, it is sufficient to compute r to determine a(n). - Michael De Vlieger, Oct 26 2015

Examples

			n=6: a(6) = gcd(720, 60, 46656) = 12.
Since only 1 and 5 are relatively prime to 6, a(6) = lcm(1,2,3,4,5,6) / lcm(1,5) = 60/5 = 12.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000 (First 1000 terms from Harry J. Smith)

Crossrefs

Cf. A000142, A000312, A051696. Equals A003418(n)/A038610(n).

Programs

GAP

List([1..70],n->Gcd(Factorial(n),n^n,Lcm([1..n]))); # Muniru A Asiru, Mar 20 2018

Maple

A064446 := n -> ilcm(seq(i,i=1..n))/ilcm(op(select(k->igcd(n,k)=1,[$1..n])));
seq(A064446(i),i=0..61); # Peter Luschny, Jun 25 2011
N:= 1000: # to get a(1) to a(N)
Primes:= select(isprime, [2,seq(2*i+1,i=1..floor((N-1)/2))]):
A:= Vector(N,1):
for p in Primes do
  for d from 1 to floor(log[p](N)) do
    for j from p^d to min(N, p^(d+1)-p) by p do
       A[j]:= A[j]*p^d
od od od:
convert(A,list); # Robert Israel, Oct 26 2015

Mathematica

Table[GCD[n!,n^n,LCM@@Range[n]],{n,70}] (* Harvey P. Dale, Jun 25 2011 *)
f[n_] := Block[{p = First /@ FactorInteger@ n}, Times @@ Power @@@ Transpose[{p, Floor@ Log[#, n] & /@ p}]]; {1}~Join~Table[f@ n, {n, 2, 10000}] (* Michael De Vlieger, Oct 26 2015 *)

PARI

L=1; for (n=1, 1000, L=lcm(L, n); write("b064446.txt", n, " ", gcd(n^n, L))) \\ Harry J. Smith, Sep 14 2009

PARI

a(n)=my(f=factor(n)); for(i=1,#f~, f[i,2]=logint(n,f[i,1])); factorback(f) \\ Charles R Greathouse IV, Nov 19 2015

PARI

a(n) = gcd(n^n, lcm(vector(n, k, k))); \\ Michel Marcus, Mar 18 2018

Formula

a(n) = gcd(A000142(n), A000312(n), A003418(n)) = gcd(A000312(n), A003418(n)) = gcd(A051696(n), A003418(n)).

a(n) = Product_{prime p | n} p^floor(log_p(n)). - Robert Israel, Oct 26 2015

a(n) = e^(Sum_{k=1..n} (floor(n^k/k) - floor((n^k - 1)/k))*Mangoldt(k)) where Mangoldt is the Mangoldt function. - Anthony Browne, Jun 16 2016