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%I #25 Nov 23 2021 09:56:18
%S 1,1,1,1,2,2,12,12,48,144,1440,1440,34560,34560,483840,7257600,
%T 58060800,58060800,3135283200,3135283200,125411328000,2633637888000,
%U 57940033536000,57940033536000,5562243219456000,27811216097280000,723091618529280000,6507824566763520000
%N a(n) = exp(-Sum_{k=1..n} Sum_{d|k, d prime} moebius(d)*log(k/d)).
%C The author proposes to denote this sequence lcm_{p}(n) as lcm(n) = lcm({1,2,..n}) = exp(Sum_{k=1..n} Sum_{d|k} moebius(d)*log(k/d)).
%C For n > 0 the a(n) are the partial products of A205959(n), which is the exponential of a modified von Mangoldt function where the divisors are restricted to prime divisors.
%H Michel Marcus, <a href="/A205957/b205957.txt">Table of n, a(n) for n = 0..400</a>
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/VonMangoldtTransformation">The von Mangoldt Transformation.</a>
%F a(n) = Product_{p prime, p<=n} (floor(n/p)!). - _Ridouane Oudra_, Nov 22 2021
%p with(numtheory):
%p A205957 := proc(n) simplify(exp(-add(add(mobius(d)*log(k/d), d=select(isprime, divisors(k))),k=1..n))) end: seq(A205957(i), i=0..27);
%t a[n_] := Exp[-Sum[ MoebiusMu[p] Log[k/p], {k, 1, n}, {p, FactorInteger[k][[All, 1]]}]]; Table[a[n], {n, 0, 27}] (* _Jean-François Alcover_, Jun 27 2013 *)
%o (Sage)
%o def A205957(n) : return simplify(exp(-add(add(moebius(p)*log(k/p) for p in prime_divisors(k)) for k in (1..n))))
%o (PARI) a(n)=prod(k=4,n,my(f=factor(k)[, 1]); prod(i=1, #f, k/f[i])) \\ _Charles R Greathouse IV_, Jun 27 2013
%Y Cf. A003418, A205959, A216152, A216153.
%K nonn
%O 0,5
%A _Peter Luschny_, Sep 01 2012