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

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, 42, 28, 2, 60, 4, 924, 231, 3570, 210, 6, 2, 51870, 2730, 420, 42, 2310, 330, 4620, 210, 9660, 420, 420, 210, 6630, 1326, 1716, 66, 660, 220, 1596, 114, 1740, 60, 60, 12, 1861860, 60060

Offset: 1

Author

Harald Hofstätter, Oct 07 2020

Keywords

• nonn

Comments

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.

n!*a(n) is the smallest common denominator of the n-th degree coefficients of the Baker-Campbell-Hausdorff series.

Links

• Harald Hofstätter, Table of n, a(n) for n = 1..20000
• Harald Hofstätter, Denominators of coefficients of the Baker-Campbell-Hausdorff series, arXiv:2010.03440 [math.NT], 2020.
• Harald Hofstätter, Smallest common denominators for the homogeneous components of the Baker-Campbell-Hausdorff series, arXiv:2012.03818 [math.NT], 2020.
• Harald Hofstätter, A simple and efficient algorithm for computing the Baker-Campbell-Hausdorff series, arXiv:2212.01290 [math.RA], 2022.
• Eric Weisstein's World of Mathematics, Baker-Campbell-Hausdorff Series.

Crossrefs

Cf. A007947 (squarefree kernel), A195441.

Programs

Julia

using Primes
A338025(n::Int) =
    prod([p^(floor(Int, log(p, sum(digits(n, base=p)))))
  for p in 2:n if isprime(p)])
println([A338025(n) for n = 1:50])

Maple

A338025 := n->mul(map(p->p^(ilog[p](add(i, i=convert(n, base, p)))), select(isprime, [seq(p, p=2..n)]))):
seq(A338025(n), n=1..50);

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

Formula

A007947(a(n)) = A195441(n-1).