A195441 -id:A195441 - cp's OEIS frontend

A287706 a(n) = denominator(Bernoulli(n, x/2) - Bernoulli(n, x)).

1, 2, 4, 8, 16, 96, 64, 384, 256, 2560, 1024, 6144, 4096, 860160, 81920, 98304, 65536, 1966080, 1310720, 55050240, 7340032, 346030080, 20971520, 125829120, 16777216, 9160359936, 469762048, 134217728, 268435456, 8053063680, 1073741824, 496068722688, 330712481792

Offset: 0

Peter Luschny, Aug 08 2017

nonn

Cf. A195441, A224783, A287705, A290645.

a := n -> denom(bernoulli(n, x/2)-bernoulli(n, x)): seq(a(i), i=0..32);

A290645 a(n) = denominator(Bernoulli(n, x) - Bernoulli(n, 1/2)).

Original entry on oeis.org

1, 2, 4, 2, 16, 6, 64, 6, 768, 10, 1024, 6, 4096, 210, 245760, 6, 196608, 30, 1310720, 210, 22020096, 330, 62914560, 30, 251658240, 546, 1409286144, 14, 268435456, 30, 1073741824, 462, 992137445376, 3570, 1803886264320, 6, 68719476736, 51870, 375208342978560

Offset: 0

Peter Luschny, Aug 08 2017

nonn

Cf. A195441, A224783, A287705, A287706.

a := n -> denom(bernoulli(n, x)-bernoulli(n, 1/2)): seq(a(i), i=0..38);

A338025 a(n) = LCM(kj_1!...*j_k! : j_1,...,j_k>=1, j_1+...+j_k=n, k=1,...,n)/n!.

Original entry on oeis.org

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

Harald Hofstätter, Oct 07 2020

nonn

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.

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.

Cf. A007947 (squarefree kernel), A195441.

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])

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);

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

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

A341109 a(n) = denominator(p(n, x)) / (n!denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

Original entry on oeis.org

1, 1, 2, 4, 8, 16, 96, 192, 1152, 768, 1536, 3072, 18432, 36864, 221184, 147456, 884736, 1769472, 10616832, 21233664, 637009920, 424673280, 2548039680, 5096079360, 152882380800, 61152952320, 366917713920, 81537269760, 163074539520, 326149079040, 1956894474240

Offset: 0

Peter Luschny, Feb 06 2021

nonn

The challenge is to characterize the sequence purely arithmetically, i.e., without reference to the Eulerian numbers or the Bernoulli polynomials.

András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.
J.-L. Chabert and P.-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences In: J. W. Brewer, S. Glaz, W. J. Heinzer, B. M. Olberding (eds), Multiplicative Ideal Theory in Commutative Algebra. Springer, Boston, MA., 2006.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, arXiv:1705.03857 [math.NT] 2017, Amer. Math. Monthly.

Cf. A100655, A053657 (Minkowski), A341107, A341108, A318256, A144845, A163176, A201637 (Eulerian2), A036689, A324370, A007947, A324369, A195441.

Epoly := proc(n, x) add(combinat:-eulerian2(n, k)*binomial(x+k, 2*n), k = 0..n) / mul(j-x, j = 1..n): simplify(expand(%)) end:
seq(denom(Epoly(n, x)) / (n!*denom(bernoulli(n, x))), n = 0..30);

A053657[n_] := Product[p^Sum[Floor[(n-1)/((p-1) p^k)], {k,0,n}],{p, Prime[Range[n]]}];
A144845[n_] := Denominator[Together[BernoulliB[n, x]]];
A163176[n_] := A053657[n] / n!;
Table[(n + 1) A163176[n + 1] / A144845[n], {n, 0, 30}]

def A341109(n): # uses[A341108, A318256]
    return A341108(n)//A318256(n)
print([A341109(n) for n in (0..30)])

a(n) = A053657(n+1)/(n!*A144845(n)).
a(n) = (n+1)*A163176(n+1)/A144845(n).
a(n) = A341108(n)/A318256(n).
a(n) = A341107(n)*A324369(n+1).
a(n) = A341108(n)/A324370(n+1).
a(n) = A341108(n)*A007947(n+1)/A144845(n).
a(n) = A341108(n)*A324369(n+1)/A195441(n).
prime(n) divides a(k) for k >= A036689(n).
2^(n-1) divides exactly a(n) for n >= 2.

cp's OEIS Frontend

A287706 a(n) = denominator(Bernoulli(n, x/2) - Bernoulli(n, x)).

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A290645 a(n) = denominator(Bernoulli(n, x) - Bernoulli(n, 1/2)).

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Author

Keywords

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Maple

A338025 a(n) = LCM(kj_1!...*j_k! : j_1,...,j_k>=1, j_1+...+j_k=n, k=1,...,n)/n!.

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Formula

A341109 a(n) = denominator(p(n, x)) / (n!denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

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Mathematica

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Formula

cp's OEIS Frontend

A287706 a(n) = denominator(Bernoulli(n, x/2) - Bernoulli(n, x)).

Views

Author

Keywords

Crossrefs

Programs

Maple

A290645 a(n) = denominator(Bernoulli(n, x) - Bernoulli(n, 1/2)).

Views

Author

Keywords

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Julia

Maple

PARI

Formula

A341109 a(n) = denominator(p(n, x)) / (n!*denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)*binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Sage

Formula

A338025 a(n) = LCM(kj_1!...*j_k! : j_1,...,j_k>=1, j_1+...+j_k=n, k=1,...,n)/n!.

A341109 a(n) = denominator(p(n, x)) / (n!denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.