OFFSET
1,2
COMMENTS
The title is analogous to the title of A061446.
For any integer sequence a, the sequence b such that b(n) = Product_{d|n} a(d) is a divisibility sequence. Not every divisibility sequence b corresponds to some integer sequence a such that b(n) = Product_{d|n} a(d), however.
This sequence is not itself a divisibility sequence; a(15) does not divide a(30), for example.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..456
Morgan Ward, A note on divisibility sequences, Bull. Amer. Math. Soc., 45 (1939), 334-336.
FORMULA
a(n) = Product_{i = 1..n, gcd(n, i) = 1} lcm (1..floor(n/i)).
a(n) = Product_{i = 1..floor(n/2), gcd(n, i) = 1} lcm (1..floor(n/i)) (equivalent formula).
a(n) = n! iff n is prime.
a(n) = Product_{d|n} (d!)^moebius(n/d). - Joerg Arndt, Jan 18 2015
a(n) = Product_{k=1..n} (gcd(n,k)!)^(mu(n/gcd(n,k))/phi(n/gcd(n,k))) = Product_{k=1..n} ((n/gcd(n,k))!)^(mu(gcd(n,k))/phi(n/gcd(n,k))) where mu = A008683, phi = A000010. - Richard L. Ollerton, Nov 08 2021
EXAMPLE
The divisors of 10 are 1, 2, 5 and 10. 10! = a(1) * a(2) * a(5) * a(10) = 1 * 2 * 120 * 15120 = 3628800.
Between 1 and 10 inclusive, 4 integers are coprime to 10: 1, 3, 7 and 9. Let b(n) = lcm (1...n) = A003418(n), and let [x] denote the floor function. Then:
a(10) = b[10/1] * b[10/3] * b[10/7] * b[10/9]
" " = b(10) * b(3) * b(1) * b(1)
" " = 2520 * 6 * 1 * 1
" " = 15120.
MATHEMATICA
Array[Product[(d!)^MoebiusMu[#/d], {d, Divisors[#]}] &, 24] (* Michael De Vlieger, Nov 11 2021 *)
PROG
(PARI) a(n)={my(r=1); fordiv(n, d, r*=d!^moebius(n/d)); r} \\ Joerg Arndt, Jan 18 2015
CROSSREFS
KEYWORD
nonn
AUTHOR
Matthew Vandermast, Dec 16 2014
STATUS
approved