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Primitive part of n! (for n>=1): n! = Product_{d|n} a(d).
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%I #29 Nov 27 2021 20:04:22

%S 1,2,6,12,120,60,5040,1680,60480,15120,39916800,55440,6227020800,

%T 8648640,1816214400,518918400,355687428096000,147026880,

%U 121645100408832000,55870214400,1689515283456000,14079294028800,25852016738884976640000,771008958720

%N Primitive part of n! (for n>=1): n! = Product_{d|n} a(d).

%C The title is analogous to the title of A061446.

%C 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.

%C This sequence is not itself a divisibility sequence; a(15) does not divide a(30), for example.

%H Michael De Vlieger, <a href="/A250269/b250269.txt">Table of n, a(n) for n = 1..456</a>

%H Morgan Ward, <a href="http://dx.doi.org/10.1090/S0002-9904-1939-06980-2">A note on divisibility sequences</a>, Bull. Amer. Math. Soc., 45 (1939), 334-336.

%F a(n) = Product_{i = 1..n, gcd(n, i) = 1} lcm (1..floor(n/i)).

%F a(n) = Product_{i = 1..floor(n/2), gcd(n, i) = 1} lcm (1..floor(n/i)) (equivalent formula).

%F a(n) = n! iff n is prime.

%F a(n) = Product_{d|n} (d!)^moebius(n/d). - _Joerg Arndt_, Jan 18 2015

%F 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

%e The divisors of 10 are 1, 2, 5 and 10. 10! = a(1) * a(2) * a(5) * a(10) = 1 * 2 * 120 * 15120 = 3628800.

%e 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:

%e a(10) = b[10/1] * b[10/3] * b[10/7] * b[10/9]

%e " " = b(10) * b(3) * b(1) * b(1)

%e " " = 2520 * 6 * 1 * 1

%e " " = 15120.

%t Array[Product[(d!)^MoebiusMu[#/d], {d, Divisors[#]}] &, 24] (* _Michael De Vlieger_, Nov 11 2021 *)

%o (PARI) a(n)={my(r=1);fordiv(n,d,r*=d!^moebius(n/d));r} \\ _Joerg Arndt_, Jan 18 2015

%Y Cf. A000142, A075071. Subsequence of A250270.

%Y Cf. A000010 (comments on product formulas), A008683.

%K nonn

%O 1,2

%A _Matthew Vandermast_, Dec 16 2014