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A250269
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Primitive part of n! (for n>=1): n! = Product_{d|n} a(d).
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4
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1, 2, 6, 12, 120, 60, 5040, 1680, 60480, 15120, 39916800, 55440, 6227020800, 8648640, 1816214400, 518918400, 355687428096000, 147026880, 121645100408832000, 55870214400, 1689515283456000, 14079294028800, 25852016738884976640000, 771008958720
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MATHEMATICA
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Array[Product[(d!)^MoebiusMu[#/d], {d, Divisors[#]}] &, 24] (* Michael De Vlieger, Nov 11 2021 *)
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PROG
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(PARI) a(n)={my(r=1); fordiv(n, d, r*=d!^moebius(n/d)); r} \\ Joerg Arndt, Jan 18 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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