A144845 -id:A144845 - cp's OEIS frontend

A285863 Numerators of Bernoulli numbers 3^n*B(n), with B(n) = A027641(n)/A027642(n).

1, -3, 3, 0, -27, 0, 243, 0, -2187, 0, 98415, 0, -122408577, 0, 11160261, 0, -51899996619, 0, 5664991530321, 0, -202943637014337, 0, 8938507796555139, 0, -22252066887294301257, 0, 7246946747292751629, 0, -181103830292539169071623

Offset: 0

Wolfdieter Lang, Apr 29 2017

The denominators are given in A285068.
In general the numbers B(d;n) = d^n*B(n), for n >= 0, have e.g.f. d*x/(exp(d*x) - 1). They are also the exponential convolution of the generalized Bernoulli numbers B[d,a](n), obtained from the generalized Stirling2 numbers S2[d,a], with the sequence {(-a)^n}_{n>=0}. See a comment in A157817 for the  B[4,1] and B[4,3] examples.
These numbers B(d;n) and their polynomials B(d;n,x) = Sum_{m=0..n} binomial(n, m)*B(d;n-m)*x^m are used in the generalized so-called Faulhaber formula for the sums of powers of arithmetic progressions defined by SP(d,a;n,m) := Sum_{j=0..m} (a + d*j)^n = Sum_{k=0..n} binomial(n, k)*a^(n-k)*d^k*SP(k,m) with SP(k,m) = SP(1,0;k,m), n >= 0, m >= 0, and 0^0 := 1.
The Faulhaber formula is: SP(d,a;n,m) = (1/(d*(n+1)))*[B(d;n+1,x = a+d*(m+1)) - B(d;n+1,x = d) - B(d;n+1,x = a)  + B(d;n+1,x=0) + d^(n+1)*[n=0]]. Here [n=0] is the Kronecker delta_{n,0} symbol: 1 if n=0 and 0 otherwise.
A simpler version of the Faulhaber formula is for a=0: SP(d,0;0,m) = m+1 and  SP(d,0;n,m) = d^n*(1/(n+1))*(B(n+1, x = m+1) - B(n+1, x=1)) for n >= 1, and for a an integer >= 1: Sum_{k=0..n} binomial(n, k)*a^(n-k) * d^k * (1/(k+1)) * (B(k+1, x=m+1) - B(k+1, x=1)). Here B(n, x) =  B(1;n,x) are the usual Bernoulli polynomials from A196838/A196839 or A053382/A053383.

Indranil Ghosh, Table of n, a(n) for n = 0..300
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.

Cf. A027641/A027642, A144845, A285068.

seq(numer(3^n*bernoulli(n)), n=0..28); # Peter Luschny, Jul 17 2017

Table[Numerator[3^n*BernoulliB[n]], {n, 0, 100}] (* Indranil Ghosh, Jul 18 2017 *)

a(n) = numerator(3^n * bernfrac(n)); \\ Ruud H.G. van Tol, Jan 31 2024

from sympy import bernoulli
def a(n): return -3 if n == 1 else (3**n * bernoulli(n)).numerator
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 18 2017

a(n) = numerator(r(n)) with r(n) = 3^n*A027641(n)/A027642(n), n >= 0.

E.g.f. {r(n)}_{n>=0}: 3*x/(exp(3*x) - 1).

A335949 a(n) = denominator(b_n(x)), where b_n(x) are the polynomials defined in A335947.

Original entry on oeis.org

1, 1, 12, 4, 240, 48, 1344, 192, 3840, 1280, 33792, 3072, 5591040, 430080, 245760, 49152, 16711680, 983040, 522977280, 27525120, 1211105280, 173015040, 1447034880, 62914560, 22900899840, 4580179968, 1409286144, 469762048, 116769423360, 4026531840, 7689065201664

Offset: 0

Peter Luschny, Jul 01 2020

nonn

The sequence can also be computed without reference to the Bernoulli polynomials (ultimately thanks to the von Staudt-Clausen theorem) by the method of Kellner and Sondow (2019). Compare the SageMath program.

Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, arXiv:1902.10672 [math.NT], 2019.

Cf. A335947/A335948, A144845, A158302.

def A335949(n):
    a = set(prime_divisors(n + 1)) - set([2])
    b = (
        p
        for p in prime_range(3, (n + 2) // (2 + n % 2))
        if not p.divides(n + 1) and sum((n + 1).digits(base=p)) >= p
    )
    p = list(a.union(set(b)))
    return 4 ^ (n // 2) * mul(p)
print([A335949(n) for n in range(31)])

a(n) = min {m | m*([x^k] b(n, x)) is an integer for all k = 0..n}.
The odd part of a(n) is squarefree (A000265).
a(n) and A144845(n) have the same odd prime factors.
a(n)/A144845(n) = 4^floor(n/2)/2 for n >= 1.
a(n)/rad(a(n)) = A158302(n+1), (rad=A007947).

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.

A366572 a(n) = denominator(Bernoulli(n, x)) / denominator(Bernoulli''(n, x)).

Original entry on oeis.org

1, 2, 6, 2, 30, 6, 42, 6, 10, 10, 66, 6, 2730, 210, 2, 6, 510, 10, 798, 42, 110, 330, 138, 2, 546, 546, 2, 2, 870, 30, 14322, 462, 170, 510, 6, 2, 1919190, 51870, 2, 6, 13530, 110, 1806, 42, 46, 690, 1410, 2, 1326, 1326, 22, 66, 1590, 10, 798, 798, 290, 870

Offset: 0

Peter Luschny, Oct 13 2023

nonn

Cf. A144845/A366168, A366571, A144845/A324370 (1st derivative).

seq(denom(bernoulli(n, x))/denom(diff(diff(bernoulli(n, x), x),x)), n=0..100);

a(n) = lcm(apply(denominator, Vec(bernpol(n))))/lcm(apply(denominator, Vec(deriv(deriv(bernpol(n)))))); \\ Michel Marcus, Oct 14 2023

a(n) = A144845(n) / A366168(n).

A319084 Numbers k such that the denominator of the Bernoulli polynomial B_k(x) is the squarefree kernel of k+1.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 27, 29, 35, 59

Offset: 1

Peter Luschny, Sep 12 2018

See A318256 for some background information.

Cf. A007947, A144845, A318256.

Numbers k such that A144845(k) / A007947(k+1) = 1.

A366571 a(n) = denominator(Bernoulli(n, x)) / denominator(Bernoulli'(n, x)).

Original entry on oeis.org

1, 2, 6, 1, 30, 1, 42, 1, 10, 1, 66, 1, 2730, 1, 2, 3, 170, 1, 798, 1, 110, 3, 46, 1, 546, 1, 2, 1, 870, 1, 14322, 1, 170, 3, 2, 1, 1919190, 1, 2, 3, 4510, 1, 1806, 1, 46, 15, 94, 1, 1326, 1, 22, 3, 530, 1, 798, 1, 290, 3, 118, 1, 56786730, 1, 2, 3, 34, 5, 64722

Offset: 0

Peter Luschny, Oct 13 2023

Cf. A144845/A324370, A366572, A144845/A366168 (2nd derivative).

seq(denom(bernoulli(n, x))/denom(diff(bernoulli(n, x), x)), n = 0..66);

a(n) = lcm(apply(denominator, Vec(bernpol(n))))/lcm(apply(denominator, Vec(deriv(bernpol(n))))); \\ Michel Marcus, Oct 14 2023

a(n) = A144845(n) / A324370(n).

A366573 a(n) = denominator(Bernoulli'(n, x)) / denominator(Bernoulli''(n, x)).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 6, 1, 10, 1, 6, 1, 210, 1, 2, 3, 10, 1, 42, 1, 110, 3, 2, 1, 546, 1, 2, 1, 30, 1, 462, 1, 170, 3, 2, 1, 51870, 1, 2, 3, 110, 1, 42, 1, 46, 15, 2, 1, 1326, 1, 22, 3, 10, 1, 798, 1, 290, 3, 2, 1, 930930, 1, 2, 3, 34, 5, 966, 1, 2, 3, 110, 1

Offset: 0

Peter Luschny, Oct 13 2023

nonn

Cf. A324370/A366168, A366572, A144845/A366168.

seq(denom(diff(bernoulli(n, x), x))/denom(diff(diff(bernoulli(n, x), x),x)), n = 0..100);

a(n) = lcm(apply(denominator, Vec(deriv(bernpol(n)))))/ lcm(apply(denominator, Vec(deriv(deriv(bernpol(n)))))); \\ Michel Marcus, Oct 14 2023

a(n) = A324370(n) / A366168(n).

cp's OEIS Frontend

A285863 Numerators of Bernoulli numbers 3^n*B(n), with B(n) = A027641(n)/A027642(n).

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A335949 a(n) = denominator(b_n(x)), where b_n(x) are the polynomials defined in A335947.

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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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Maple

Mathematica

Sage

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A366572 a(n) = denominator(Bernoulli(n, x)) / denominator(Bernoulli''(n, x)).

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Maple

PARI

Formula

A319084 Numbers k such that the denominator of the Bernoulli polynomial B_k(x) is the squarefree kernel of k+1.

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Comments

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Formula

A366571 a(n) = denominator(Bernoulli(n, x)) / denominator(Bernoulli'(n, x)).

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Programs

Maple

PARI

Formula

A366573 a(n) = denominator(Bernoulli'(n, x)) / denominator(Bernoulli''(n, x)).

Views

Author

Keywords

Crossrefs

Programs

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.