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A338025 a(n) = LCM(k*j_1!*...*j_k! : j_1,...,j_k>=1, j_1+...+j_k=n, k=1,...,n)/n!.

%I A338025 #46 Dec 20 2022 01:37:49
%S A338025 1,1,2,1,6,2,6,3,10,2,6,2,210,30,12,3,30,10,210,42,330,30,60,30,546,
%T A338025 42,28,2,60,4,924,231,3570,210,6,2,51870,2730,420,42,2310,330,4620,
%U A338025 210,9660,420,420,210,6630,1326,1716,66,660,220,1596,114,1740,60,60,12,1861860,60060
%N A338025 a(n) = LCM(k*j_1!*...*j_k! : j_1,...,j_k>=1, j_1+...+j_k=n, k=1,...,n)/n!.
%C A338025 For each prime p >= 2, the exponent of p in a(n) is the largest integer t such that p^t is less than or equal to the sum of digits of n in base p.
%C A338025 n!*a(n) is the smallest common denominator of the n-th degree coefficients of the Baker-Campbell-Hausdorff series.
%H A338025 Harald Hofstätter, <a href="/A338025/b338025.txt">Table of n, a(n) for n = 1..20000</a>
%H A338025 Harald Hofstätter, <a href="https://arxiv.org/abs/2010.03440">Denominators of coefficients of the Baker-Campbell-Hausdorff series</a>, arXiv:2010.03440 [math.NT], 2020.
%H A338025 Harald Hofstätter, <a href="https://arxiv.org/abs/2012.03818">Smallest common denominators for the homogeneous components of the Baker-Campbell-Hausdorff series</a>, arXiv:2012.03818 [math.NT], 2020.
%H A338025 Harald Hofstätter, <a href="https://arxiv.org/abs/2212.01290">A simple and efficient algorithm for computing the Baker-Campbell-Hausdorff series</a>, arXiv:2212.01290 [math.RA], 2022.
%H A338025 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Baker-Campbell-HausdorffSeries.html">Baker-Campbell-Hausdorff Series</a>.
%F A338025 A007947(a(n)) = A195441(n-1).
%p A338025 A338025 := n->mul(map(p->p^(ilog[p](add(i, i=convert(n, base, p)))), select(isprime, [seq(p, p=2..n)]))):
%p A338025 seq(A338025(n), n=1..50);
%o A338025 (Julia)
%o A338025 using Primes
%o A338025 A338025(n::Int) =
%o A338025     prod([p^(floor(Int, log(p, sum(digits(n, base=p)))))
%o A338025   for p in 2:n if isprime(p)])
%o A338025 println([A338025(n) for n = 1:50])
%o A338025 (PARI) a(n) = {my(v = matrix(primepi(n), 2, i, j, my(p=prime(i)); if (j==1, p, logint(sumdigits(n, p), p)))); factorback(v);} \\ _Michel Marcus_, Oct 08 2020
%Y A338025 Cf. A007947 (squarefree kernel), A195441.
%K A338025 nonn
%O A338025 1,3
%A A338025 _Harald Hofstätter_, Oct 07 2020