A286517 -id:A286517 - cp's OEIS frontend

A144845 Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers.

1, 2, 6, 2, 30, 6, 42, 6, 30, 10, 66, 6, 2730, 210, 30, 6, 510, 30, 3990, 210, 2310, 330, 690, 30, 2730, 546, 42, 14, 870, 30, 14322, 462, 39270, 3570, 210, 6, 1919190, 51870, 2730, 210, 94710, 2310, 99330, 2310, 4830, 4830, 9870, 210, 46410, 6630, 14586, 858

Offset: 0

T. D. Noe, Sep 22 2008

nonn

The lcm of the terms in row n of A053383. It appears that the Bernoulli polynomial B(n,x) is irreducible for all even n.
This sequence appears in a paper of Bazsó & Mező, who use this sequence to give necessary and sufficient conditions for power sums to be integer polynomials. - Istvan Mezo, Mar 20 2016
In "The denominators of power sums of arithmetic progressions" Corollary 1, we give a simple way to compute a(n) without using Bernoulli polynomials. Namely, a(n) equals (product of the primes dividing n+1) times (product of the primes p <= (n+1)/(2+(n+1 mod 2)) not dividing n+1 such that the sum of the base-p digits of n+1 is at least p). - Bernd C. Kellner and Jonathan Sondow, May 15 2017

Bernd C. Kellner, Table of n, a(n) for n = 0..10000 (n = 0..1000 from T. D. Noe)
András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.
András Bazsó and István Mező, Some Notes on Alternating Power Sums of Arithmetic Progressions, J. Int. Seq., Vol. 21 (2018), Article 18.7.8.
Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
Bernd C. Kellner, On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients, J. Integer Seq. 27 (2024), Article 24.2.8, 11 pp.; arXiv:2310.01325 [math.NT], 2023.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Bernoulli Polynomial.

Cf. A027642, A053383, A064538, A195441, A286515, A286516, A286517, A324369, A324370, A324371.

seq(denom(bernoulli(i,x)),i=0..51); # Peter Luschny, Jun 16 2012

(* From Bernd C. Kellner, Oct 18 2023: (Start) *)
(* Denominator formula *)
Table[Denominator[Together[BernoulliB[n, x]]], {n, 0, 51}]
(* Product formula *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]]; rad[n_] := Times @@ Select[Divisors[n], PrimeQ]; (* A324370 *) DD2[n_] := Times @@ Select[Prime[Range[PrimePi[(n+1)/(2+Mod[n+1, 2])]]], !Divisible[n, #] && SD[n, #] >= # &];
DB[n_] := DD2[n+1] rad[n+1]; Table[DB[n], {n, 0, 51}]
(* (End) *)

a(n) = lcm(apply(x->denominator(x), Vec(bernpol(n)))); \\ Michel Marcus, Mar 03 2020

def A144845(n):
    return mul(prime_divisors(n+1) + [p for p in (2..(n+2)//(2+n%2))
    if is_prime(p) and not p.divides(n+1) and sum((n+1).digits(base=p)) >= p])
print([A144845(n) for n in (0..51)]) # Peter Luschny, Sep 12 2018

From Bernd C. Kellner, Oct 18 2023: (Start)
Let rad(n) = A007947(n) be the radical of n. Let (n)_m be the falling factorial. Let f^(m)(x) denote the m-th derivative of f(x).
a(n) = lcm(A195441(n-1), A027642(n)) = lcm(denom(B(n,x)-B_n), denom(B_n)) = denom(B(n,x)).
a(n) = lcm(A195441(n), rad(n+1)).
a(n) = lcm(a(n+1), rad(n+1)), if n >= 2 is even.
a(2n)/a(2n+1) = A286517(n), if n >= 1.
a(n) = A324369(n+1) * A324370(n+1) * A324371(n+1).
a(n) = A324370(n+1) * rad(n+1).
a(n) = rad(A064538(n)).
If n >= m >= 1, then denom(B^(m)(n,x)) = a(n-m)/gcd(a(n-m), (n)A324370(n-m+1)/gcd(A324370(n-m+1),%20(n)">m) = A324370(n-m+1)/gcd(A324370(n-m+1), (n){m-1}).
(See papers of Kellner and Kellner & Sondow.) (End)

A286516 a(n) = b(2n-1)/b(2n) where b(n) = A195441(n-1) = denominator(Bernoulli_{n}(x) - Bernoulli_{n}).

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 1, 13, 7, 15, 2, 17, 3, 19, 5, 7, 11, 23, 1, 5, 13, 3, 7, 29, 5, 31, 2, 11, 17, 7, 1, 37, 19, 13, 5, 41, 21, 43, 11, 3, 23, 47, 1, 7, 5, 17, 13, 53, 3, 11, 7, 19, 29, 59, 1, 61, 31, 7, 2, 65, 11, 67, 17, 23, 5, 71, 1, 73, 37

Offset: 1

Bernd C. Kellner and Jonathan Sondow, May 12 2017

nonn

a(n) is an integer for all n, a(n) is odd if n is not a power of 2, a(2^k)=2 for all k>=1, a(n)=1 infinitely often, and a(n)=p infinitely often for every prime p. See Cor. 2 and Cor. 3 in "The denominators of power sums of arithmetic progressions". See also "Power-sum denominators".

G. C. Greubel, Table of n, a(n) for n = 1..1000
Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017.

Cf. A027642, A064538, A144845, A195441, A286515, A286517, A286762, A286763.

b[n_] := Denominator[ Together[ BernoulliB[n, x] - BernoulliB[n]]]; Table[
b[2 n - 1]/b[2 n], {n, 1, 74}]

a(n) = A195441(2*n-2) / A195441(2*n-1).

A318256 a(n) = (denominator of B(n,x)) / (the squarefree kernel of n+1), where B(n,x) is the n-th Bernoulli polynomial.

Original entry on oeis.org

1, 1, 2, 1, 6, 1, 6, 3, 10, 1, 6, 1, 210, 15, 2, 3, 30, 5, 210, 21, 110, 15, 30, 5, 546, 21, 14, 1, 30, 1, 462, 231, 1190, 105, 6, 1, 51870, 1365, 70, 21, 2310, 55, 2310, 105, 322, 105, 210, 35, 6630, 663, 286, 33, 330, 55, 798, 57, 290, 15, 30, 1, 930930, 15015

Offset: 0

Peter Luschny, Sep 12 2018

nonn

			a(59) = 1 because there exist no number which satisfies the definition (and the product of an empty set is 1).
a(60) = 930930 because {2, 3, 5, 7, 11, 13, 31} are the only primes which satisfy the definition.
The denominator of the Bernoulli polynomial B_n(x) equals the squarefree kernel of n+1 if n is in {0, 1, 3, 5, 9, 11, 27, 29, 35, 59}. These might be the only numbers with this property.

Peter Luschny, Table of n, a(n) for n = 0..1000
András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.
Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits, Integers 21 (2021), #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019.
Bernd C. Kellner, On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients, 9 pp.; arXiv:2310.01325 [math.NT], 2023.

a(n) = A144845(n) / A007947(n+1).
Cf. A324370 (same sequence with offset 1).
Cf. A027642, A064538, A195441, A286515, A286516, A286517, A286762, A286763, A319084.

a := n -> denom(bernoulli(n, x)) / mul(p, p in numtheory:-factorset(n+1)):
seq(a(n), n=0..61);

sfk[n_] := Times @@ FactorInteger[n][[All, 1]];
a[n_] := (BernoulliB[n, x] // Together // Denominator)/sfk[n+1];
Table[a[n], {n, 0, 61}] (* Jean-François Alcover, Feb 14 2019 *)

def A318256(n): return mul([p for p in (2..(n+2)//(2+n%2))
                if is_prime(p)
                and not p.divides(n+1)
                and sum((n+1).digits(base=p)) >= p])
print([A318256(n) for n in (0..61)])

Let Q(n) = {p <= floor((n + 2)/(2 + n mod 2)) and p is prime and p does not divide n + 1 and the sum of the digits in base p of n+1 is at least p} then a(n) = Product_{p in Q(n)} p. (See the Kellner & Sondow links.)

a(n) = denominator(Bernoulli'(n+1, x)), where ' denotes d/dx. - Peter Luschny, Oct 15 2023

A286515 a(n) = denominator(Bernoulli_{n}(x)) / denominator(Bernoulli_{n}).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 6, 1, 10, 1, 6, 1, 210, 5, 6, 1, 30, 5, 210, 7, 330, 5, 30, 1, 546, 7, 14, 1, 30, 1, 462, 77, 3570, 35, 6, 1, 51870, 455, 210, 7, 2310, 55, 2310, 7, 4830, 35, 210, 1, 6630, 221, 858, 11, 330, 55, 798, 19, 870, 5, 30, 1, 930930, 5005, 4290

Offset: 0

Bernd C. Kellner and Jonathan Sondow, May 11 2017

nonn

a(n) is a squarefree integer for all n, a(n) is odd if n>=0 is even, and a(n) is even if n>=3 is odd. See "Power-sum denominators", Thm. 4, pp. 12-13, and "The denominators of power sums of arithmetic progressions", Thm. 3, pp. 3 and 11-12.

Robert Israel, Table of n, a(n) for n = 0..5900
Bernd C. Kellner, On a product of certain primes, J. Number Theory, 179 (2017), 126-141; arXiv:1705.04303 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:1705.03857 [math.NT], 2017.
Bernd C. Kellner and Jonathan Sondow, The denominators of power sums of arithmetic progressions, Integers 18 (2018), #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017.

Cf. A027642, A064538, A144845, A195441, A286516, A286517.

seq(denom(bernoulli(n,x))/denom(bernoulli(n)), n=0..100); # Robert Israel, May 24 2017

Table[ Denominator[ Together[ BernoulliB[n, x]]]/Denominator[ BernoulliB[n]], {n, 0, 63}]

apply( a(n)=denominator(content(bernpol(n)))/denominator(bernfrac(n)), [1..50]) \\ M. F. Hasler, Dec 10 2018

a(n) = A144845(n)/A027642(n) = A195441(n-1)/gcd(A195441(n-1),A027642(n)).

cp's OEIS Frontend

A144845 Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers.

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A286516 a(n) = b(2*n-1)/b(2*n) where b(n) = A195441(n-1) = denominator(Bernoulli_{n}(x) - Bernoulli_{n}).

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A318256 a(n) = (denominator of B(n,x)) / (the squarefree kernel of n+1), where B(n,x) is the n-th Bernoulli polynomial.

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Maple

Mathematica

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Formula

A286515 a(n) = denominator(Bernoulli_{n}(x)) / denominator(Bernoulli_{n}).

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Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A286516 a(n) = b(2n-1)/b(2n) where b(n) = A195441(n-1) = denominator(Bernoulli_{n}(x) - Bernoulli_{n}).