A144845 -id:A144845 - cp's OEIS frontend

A324369 Product of all primes p dividing n such that the sum of the base p digits of n is at least p, or 1 if no such prime.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 3, 1, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 1, 3, 2, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 15, 2, 1, 6, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 1, 6, 1, 2, 3, 1, 5, 6, 1, 2, 3, 10, 1, 6, 1, 2, 3, 2, 1, 6, 1, 2, 1, 2, 1, 2, 5, 2, 3, 2, 1, 10, 7, 2, 3, 2, 5, 6, 1

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 24 2019

a(n) = n iff n divides denominator(Bernoulli_n(x) - Bernoulli_n) (see A195441).
a(n) = n iff n = 1 or n is in A324315.
a(n) = n if n is a Carmichael number (A002997).
See the section on Bernoulli polynomials in Kellner and Sondow 2019.

			6 = 2 * 3, and 6 = 110_2 in base 2 with 1+1+0 >= 2, but 6 = 20_3 in base 3 with 2+0 = 2 < 3, so a(6) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000
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, 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.

Cf. A007947, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324370, A324371, A324404, A324405.

g:= proc(n,p) convert(convert(n,base,p),`+`) >= p end proc:
f:= proc(n) local p;
      convert(select(p -> g(n,p), numtheory:-factorset(n)),`*`)
end proc:
map(f, [$1..100]); # Robert Israel, Feb 28 2019

SD[n_, p_] := If[n < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
DD1[n_] := Times @@ Select[LP[n], SD[n, #] >= # &];
Table[DD1[n], {n, 1, 100}]

from math import prod
from sympy.ntheory import digits
from sympy import primefactors as pf
def a(n): return prod(p for p in pf(n) if sum(digits(n, p)[1:]) >= p)
print([a(n) for n in range(1, 98)]) # Michael S. Branicky, Jul 03 2022

a(n) * A324371(n) = A007947(n) = radical(n).
a(n) * A324370(n) = A195441(n-1) = denominator(Bernoulli_n(x) - Bernoulli_n).
a(n) * A324370(n) * A324371(n) = A144845(n-1) = denominator(Bernoulli_{n-1}(x)).

A324371 Product of all primes p dividing n such that the sum of the base p digits of n is less than p, or 1 if no such prime.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 3, 5, 11, 3, 13, 7, 5, 2, 17, 3, 19, 5, 7, 11, 23, 1, 5, 13, 3, 7, 29, 15, 31, 2, 11, 17, 35, 3, 37, 19, 13, 5, 41, 7, 43, 11, 1, 23, 47, 1, 7, 5, 17, 13, 53, 3, 55, 7, 19, 29, 59, 5, 61, 31, 7, 2, 13, 11, 67, 17, 23, 7, 71, 1, 73, 37, 5, 19, 77, 13, 79, 5, 3, 41, 83, 21

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 25 2019

Does not contain any elements of A324315, and thus none of the Carmichael numbers A002997.

See the section on Bernoulli polynomials in Kellner and Sondow 2019.

			For p = 2 and 3, the sum of the base p digits of 6 is 1+1+0 = 2 >= 2 and 2+0 = 2 < 3, respectively, so a(6) = 3.

Robert Israel, Table of n, a(n) for n = 1..10000
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, 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.

Cf. A002997, A007947, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324404, A324405.

f:= n -> convert(select(p -> convert(convert(n,base,p),`+`)Robert Israel, Apr 26 2020

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
DD3[n_] := Times @@ Select[LP[n], SD[n, #] < # &];
Table[DD3[n], {n, 1, 100}]

from math import prod
from sympy.ntheory import digits
from sympy import primefactors as pf
def a(n): return prod(p for p in pf(n) if sum(digits(n, p)[1:]) < p)
print([a(n) for n in range(1, 85)]) # Michael S. Branicky, Jul 03 2022

a(n) * A324369(n) = A007947(n) = radical(n).

a(n) * A195441(n) = a(n) * A324369(n) * A324370(n) = A144845(n-1) = denominator(Bernoulli_{n-1}(x)).

A094960 Positive integers k such that the derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.

Original entry on oeis.org

1, 2, 4, 6, 10, 12, 28, 30, 36, 60

Offset: 1

Benoit Cloitre, Jun 19 2004

From Max Alekseyev, Dec 08 2011: (Start)
There are no other terms below 10^9.
k belongs to this sequence if k*binomial(k-1,m)*Bernoulli(m) is an integer for each m in 0..k-1. (End)
From Max Alekseyev, Jun 04 2012: (Start)
If for a prime p >= 3, k ends with base-p digits a,b with a+b >= p, then for m = (a+1)*(p-1), the number k*binomial(k-1,m)*Bernoulli(m) is not an integer (it contains p in the denominator). For p=3, this implies that k == 5, 7, or 8 (mod 9) are not in this sequence; for p=5, this implies that k == 9, 13, 14, 17, 18, 19, 21, 22, 23, or 24 (mod 25) are not in this sequence; and so on.
Conjecture: for every integer k > 78, there exists a prime p >= 3 such that the sum of last two base-p digits of k is at least p. This conjecture would imply that this sequence is finite and 60 is the last term. (End)
The conjecture is true for all k such that k+1 is not a prime, a power of 2, or a Giuga number (A007850). In this case, there exists a prime p >= 3 such that the base-p representation of k ends in a,p-1 with a > 0. - Max Alekseyev, Feb 16 2021
The sequence is finite and is a subsequence of A366169. The terms are those numbers k where A324370(k) = 1. It remains to show that 60 is the last term. This is very likely, since the terms depend on the estimation of a product of primes satisfying certain p-adic conditions that is connected with A324370. A proven asymptotic formula related to that product implies that this sequence is finite. See Kellner 2017, 2023, and BLMS 2018. - Bernd C. Kellner, Oct 02 2023

			B(6,x) = x^6 - 3*x^5 + (5/2)*x^4 - (1/2)*x^2 + 1/42 so B'(6,x) contains only integer coefficients and 6 is in the sequence.

Olivier Bordellès, Florian Luca, Pieter Moree, and Igor E. Shparlinski, Denominators of Bernoulli polynomials, Mathematika 64 (2018), 519-541.
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.

Cf. A094960, A144845, A160014, A195441, A324370, A366168, A366169.

p := n -> if denom(diff(bernoulli(n, x), x)) = 1 then n else fi:
seq(p(n), n=1..100); # Emeric Deutsch

(* From Bernd C. Kellner, Oct 02 2023. (Start) *)
(* k-th derivative of BP: *)
k = 1; Select[Range[1000], Denominator[Together[D[BernoulliB[#, x],{x, k}]]] == 1&]
(* Exact denominator formula: *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]];
DBP[n_, k_] := Module[{m = n-k+1, fac = FactorialPower[n, k]}, If[n < 1 || k < 1 || n <= k, Return[1]]; Times@@Select[Prime[Range[PrimePi[(m+1)/(2 + Mod[m+1, 2])]]], !Divisible[fac, #] && SD[m, #] >= #&]];
k = 1; Select[Range[1000], DBP[#, k] == 1&]
(* End *)

is_A094960(k) = !#select(x->(denominator(x)!=1), Vec(deriv(bernpol(k)))); \\ Michel Marcus, Feb 15 2021

from itertools import count, islice
from sympy import Poly, diff, bernoulli
from sympy.abc import x
def A094960_gen(): # generator of terms
    return filter(lambda k:k<=1 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x)).coeffs()),count(1))
A094960_list = list(islice(A094960_gen(),10)) # Chai Wah Wu, Oct 03 2023

k is a term if A324370(k) = 1. - Bernd C. Kellner, Oct 02 2023

k is a term <=> 0 = Sum_{j=0..k-1} k*binomial(k - 1, j) mod Clausen(j), where Clausen(n) = A160014(n, 1). - Peter Luschny, Oct 04 2023

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).

A366168 Denominator of the second derivative of the n-th Bernoulli polynomial B(n,x).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 15, 1, 1, 3, 5, 5, 21, 1, 5, 15, 5, 1, 21, 7, 1, 1, 1, 1, 231, 7, 35, 3, 1, 1, 1365, 35, 7, 21, 55, 55, 105, 7, 7, 105, 35, 5, 663, 13, 11, 33, 55, 1, 57, 1, 5, 15, 1, 1, 15015, 715, 715, 33, 17, 85, 2415, 35, 1, 3, 55, 55, 285285, 19019, 1001

Offset: 1

Bernd C. Kellner, Oct 02 2023

The sequence consists only of odd numbers. The denominators are connected with A324370, from which an explicit formula follows as given below. See Kellner 2023.

			B(5,x) = x^5 - (5x^4)/2 + (5 x^3)/3 - x/6 and B''(5,x) = 20x^3 - 30x^2 + 10x, so a(5) = 1.
a(14) = A324370(13)/gcd(A324370(13), 14) = 210/gcd(210, 14) = 15.

Bernd C. Kellner, Table of n, a(n) for n = 1..10000
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.

Cf. A094960, A144845, A195441, A324370, A366169.

(* k-th derivative of BP *)
k = 2; Table[Denominator[Together[D[BernoulliB[n, x], {x, k}]]], {n, 1, 100}]
(* exact denominator formula *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]];
DBP[n_, k_] := Module[{m = n-k+1, fac = FactorialPower[n, k]}, If[n < 1 || k < 1 || n <= k, Return[1]]; Times@@Select[Prime[Range[PrimePi[(m+1)/(2 + Mod[m+1, 2])]]], !Divisible[fac, #] && SD[m, #] >= #&]];
k = 2; Table[DBP[n, k], {n, 1, 100}]

from math import lcm
from sympy import Poly, diff, bernoulli
from sympy.abc import x
def A366168(n): return lcm(*(c.q for c in Poly(diff(bernoulli(n,x),x,2)).coeffs())) if n>=3 else 1 # Chai Wah Wu, Oct 04 2023

Let (n)_k be the falling factorial. Let s_p(n) be the sum of the p-adic digits of n.

a(1) = 1, and for n > 1, a(n) = A324370(n-1)/gcd(A324370(n-1), n) = Product_{prime p <= n/(2+(n mod 2)): gcd(p,(n)_2)=1, s_p(n-1) >= p} p.

A285068 Denominators of the generalized Bernoulli numbers B[3,1] = 3^n*B(n, 1/3).

Original entry on oeis.org

1, 2, 2, 1, 10, 1, 14, 1, 10, 1, 22, 1, 910, 1, 2, 1, 170, 1, 266, 1, 110, 1, 46, 1, 910, 1, 2, 1, 290, 1, 4774, 1, 170, 1, 2, 1, 639730, 1, 2, 1, 4510, 1, 602, 1, 230, 1, 94, 1, 15470, 1, 22

Offset: 0

Wolfdieter Lang, Apr 28 2017

The numerators are given in A157799.
Because B(n, 2/3) = (-1)^n*B(n, 1/3) (from the e.g.f. z*exp(x*z)/(exp(z)-1) of Bernoulli polynomials {B(n, x)}_{n>=0}) one has for the numbers B[3,2](n) = 3^n*B(n, 2/3) the numerators (-1)^n*A157799(n) and the denominators a(n).
This sequence gives also the denominators of {3^n*B(n)}_{n>=0} with numerators given in A285863.

			The Bernoulli numbers r(n) = B[3,1](n) begin: 1, -1/2, -1/2, 1, 13/10, -5, -121/14, 49, 1093/10, -809, -49205/22, 20317, 61203943/910, -722813, -5580127/2, 34607305, ...
The Bernoulli numbers B[3,2](n) begin: 1, 1/2, -1/2, -1, 13/10, 5, -121/14, -49, 1093/10, 809, -49205/22, -20317, 61203943/910, 722813, -5580127/2, -34607305, ...
From _Peter Luschny_, Mar 26 2021: (Start)
The generalized Bernoulli numbers as given in the Luschny link are different.
1, -7/2, 23/2, -35, 973/10, -245, 7943/14, -1295, 31813/10, -7721, 288715/22, -13475, 128296423/910, -882557, -4891999/2, 33870025, ...
The numerators of these numbers are in A157811. (End)

Indranil Ghosh, Table of n, a(n) for n = 0..500
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
Peter Luschny, Generalized Bernoulli numbers.

Cf. A053382/A053383, A157799, A157800, A196838/A196839, A282629, A284861, A285863.

Cf. A144845, A157811.

Table[Denominator[3^n*BernoulliB[n, 1/3]], {n, 0, 100}] (* Indranil Ghosh, Jul 18 2017 *)

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

from sympy import bernoulli, Rational
def a(n):
    return (3**n * bernoulli(n, Rational(1,3))).as_numer_denom()[1]
print([a(n) for n in range(101)])  # Indranil Ghosh, Jul 18 2017

# uses [gen_bernoulli_number from A157811]
print([denominator((-1)^n*gen_bernoulli_number(n, 3)) for n in range(23)])
# Peter Luschny, Mar 26 2021

a(n) = denominator(r(n)) with the rationals (in lowest terms) r(n) = Sum_{k=0..n} ((-1)^k / (k+1))*A282629(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A284861(n, k). r(n) = B[3,1](n) = 3^n*B(n, 1/3) with the Bernoulli polynomials A196838/A196839 or A053382/A053383.

a(n) = A157800(n)/3^n, n >= 0.

A366169 Positive integers k such that the second derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 15, 16, 21, 25, 28, 29, 30, 31, 36, 37, 55, 57, 60, 61, 70, 121, 190

Offset: 1

Bernd C. Kellner, Oct 02 2023

The sequence is finite and is a supersequence of A094960. The terms are those numbers k where the denominator A366168(k) = 1. It remains to show that 190 is the last term. This is very likely, since the terms depend on the estimation of a product of primes satisfying certain p-adic conditions that is connected with A324370. A proven asymptotic formula related to that product implies that this sequence is finite. See Kellner 2017, 2023, and BLMS 2018.

			B(5,x) = x^5 - (5x^4)/2 + (5 x^3)/3 - x/6 and B''(5,x) = 20x^3 - 30x^2 + 10x, so 5 is a term.

Olivier Bordellès, Florian Luca, Pieter Moree, and Igor E. Shparlinski, Denominators of Bernoulli polynomials, Mathematika 64 (2018), 519-541.
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.

Cf. A094960, A144845, A195441, A324370, A366168.

aList := len -> select(n -> denom(diff(diff(bernoulli(n, x), x), x)) = 1, [seq(1..len)]): aList(200);  # Peter Luschny, Oct 03 2023

(* k-th derivative of BP *)
k = 2; Select[Range[1000], Denominator[Together[D[BernoulliB[#, x],{x, k}]]] == 1&]
(* exact denominator formula *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]];
DBP[n_, k_] := Module[{m = n-k+1, fac = FactorialPower[n, k]}, If[n < 1 || k < 1 || n <= k, Return[1]]; Times@@Select[Prime[Range[PrimePi[(m+1)/(2 + Mod[m+1, 2])]]], !Divisible[fac, #] && SD[m, #] >= #&]];
k = 2; Select[Range[1000], DBP[#, k] == 1&]

isok(k) = #select(x->denominator(x)>1, Vec(deriv(deriv(bernpol(k))))) == 0; \\ Michel Marcus, Oct 03 2023

from itertools import count, islice
from sympy import Poly, diff, bernoulli
from sympy.abc import x
def A366169_gen(): # generator of terms
    return filter(lambda k:k<=2 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x,2)).coeffs()),count(1))
A366169_list = list(islice(A366169_gen(),20)) # Chai Wah Wu, Oct 03 2023

k is a term if A366168(k) = 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)).

A286517 a(n) = b(2n)/b(2n+1) where b(n) = denominator(Bernoulli_{n}(x)).

Original entry on oeis.org

3, 5, 7, 3, 11, 13, 5, 17, 19, 7, 23, 5, 3, 29, 31, 11, 35, 37, 13, 41, 43, 1, 47, 7, 17, 53, 55, 19, 59, 61, 7, 13, 67, 23, 71, 73, 5, 77, 79, 3, 83, 17, 29, 89, 13, 31, 19, 97, 11, 101, 103, 7, 107, 109, 37, 113, 23, 13, 119, 11, 41, 5, 127, 43, 131, 19, 5

Offset: 1

Bernd C. Kellner and Jonathan Sondow, May 12 2017

nonn

a(n) is an odd integer for all n, a(n)=1 infinitely often, and a(n)=p infinitely often for every odd 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, A286516.

b[n_] := Denominator[ Together[ BernoulliB[n, x]]]; Table[ b[2 n]/b[2 n + 1], {n, 1, 67}]

a(n) = A144845(2*n) / A144845(2*n+1) for n >= 1.

cp's OEIS Frontend

A324369 Product of all primes p dividing n such that the sum of the base p digits of n is at least p, or 1 if no such prime.

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A324371 Product of all primes p dividing n such that the sum of the base p digits of n is less than p, or 1 if no such prime.

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A094960 Positive integers k such that the derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.

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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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A366168 Denominator of the second derivative of the n-th Bernoulli polynomial B(n,x).

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A285068 Denominators of the generalized Bernoulli numbers B[3,1] = 3^n*B(n, 1/3).

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A366169 Positive integers k such that the second derivative of the k-th Bernoulli polynomial B(k,x) contains only integer coefficients.

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

A286517 a(n) = b(2n)/b(2n+1) where b(n) = denominator(Bernoulli_{n}(x)).