A324370 -id:A324370 - cp's OEIS frontend

A324405 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 3 (mod p-1), where s_p(m) is the sum of the base p digits of m.

3003, 3315, 5187, 7395, 8463, 14763, 19803, 26733, 31755, 47523, 50963, 58035, 62403, 88023, 105339, 106113, 123123, 139971, 152643, 157899, 166611, 178923, 183183, 191919

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 26 2019

For d >= 1 define S_d = (terms m in A324315 such that s_p(m) == d (mod p-1) if prime p divides m). Then S_1 is precisely the Carmichael numbers (A002997), S_2 is A324404, S_3 is A324405, and the union of all S_d for d >= 1 is A324315.
Subsequence of the 3-Knödel numbers (A033553). Generally, for d > 1 the terms of S_d that are greater than d form a subsequence of the d-Knödel numbers.
See Kellner and Sondow 2019.

			3003 = 3*7*11*13 is squarefree and equals 11010020_3, 11520_7, 2290_11, and 14a0_13 in base p = 3, 7, 11, and 13. Then s_3(3003) = 1+1+1+2 = 5 >= 3, s_7(3003) = 1+1+5+2 = 9 >= 7, s_11(3003) = 2+2+9 = 13 >= 11, and s_13(3003) = 1+4+a = 1+4+10 = 15 >= 13. Also, s_3(3003) = 5 == 3 (mod 2), s_7(3003) = 9 == 3 (mod 6), s_11(3003) = 13 == 3 (mod 10), and s_13(3003) = 15 == 3 (mod 12), so 3003 is a member.

Amiram Eldar, Table of n, a(n) for n = 1..2000
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), Article #A52, 21 pp.; arXiv:1902.10672 [math.NT], 2019-2021.

Cf. A002997, A033553, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324370, A324371, A324404.

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
LP[n_] := Transpose[FactorInteger[n]][[1]];
TestSd[n_, d_] := (n > 1) && (d > 0) && SquareFreeQ[n] && VectorQ[LP[n], SD[n, #] >= # && Mod[SD[n, #] - d, # - 1] == 0 &];
Select[Range[200000], TestSd[#, 3] &]

A324317 Number of primary Carmichael numbers (A324316) less than 10^n.

Original entry on oeis.org

0, 0, 0, 2, 4, 9, 19, 51, 107, 219, 417, 757, 1470, 2666, 5040, 9280, 17210, 32039, 59762, 111811, 210627, 397968

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 22 2019

The number of Carmichael numbers (A002997) less than 10^n is 0, 0, 1, 7, 16, 43, 105, 255, 646, 1547, 3605, 8241, 19279, 44706, 105212, 246683, 585355, 1401644, ... (see A055553).
The terms up to a(10) are given in Table 1 of Kellner and Sondow 2019. The terms up to a(18) and related results are given in Table 1.5 of Kellner 2019.
All computations depend on Pinch's database.

			There are two primary Carmichael numbers less than 10^4, namely, 1729 and 2821, so a(4) = 2.

Claude Goutier, Compressed text file carm10e22.gz containing all the Carmichael numbers up to 10^22.
Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, Amer. Math. Monthly, 124 (2017), 695-709; arXiv preprint, 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 preprint, arXiv:1902.10672 [math.NT], 2019-2021.
Bernd C. Kellner, On primary Carmichael numbers, Integers 22 (2022), #A38, 39 pp.; arXiv preprint, arXiv:1902.11283 [math.NT], 2019-2022.
R. G. E. Pinch, Tables relating to Carmichael numbers (The Carmichael numbers up to 10^18, 2008).
Index entries for sequences related to Carmichael numbers.

Cf. A002997, A055553, A324315, A324316, A324318, A324319, A324320, A324369, A324370, A324371, A324404, A324405.

a(11)-a(18) from Amiram Eldar, Mar 01 2019
a(19) from Amiram Eldar, Dec 05 2020
a(20)-a(22) calculated using data from Claude Goutier and added by Amiram Eldar, Apr 22 2024

A324318 Number of terms in A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) less than 10^n.

Original entry on oeis.org

0, 0, 2, 57, 636, 7048, 75150, 801931, 8350039, 86361487

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 23 2019

The number of squarefree integers less than 10^n is 0, 6, 61, 608, 6083, 60794, 607926, 6079291, 60792694, 607927124, ... (see A053462).

			There are two terms of A324315 less than 10^3, namely, 231 and 561, so a(3) = 2.

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, #A52 Integers 21 (2021), 21 pp.; arXiv:1902.10672 [math.NT], 2019.

Cf. A053462, A324315, A324316, A324317, A324319, A324320, A324369, A324370, A324371, A324404, A324405.

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

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.

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

A366186 Positive integers k such that the third 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, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 25, 26, 28, 29, 30, 31, 32, 35, 36, 37, 38, 42, 50, 52, 55, 56, 57, 58, 60, 61, 62, 66, 70, 71, 72, 78, 80, 92, 110, 121, 122, 156, 176, 177, 190, 191, 210, 392

Offset: 1

Peter Luschny, Oct 03 2023

From Bernd C. Kellner, Oct 04 2023: (Start)
As a published result on Oct 02 2023 (cf. A366169), all such sequences regarding higher derivatives of the Bernoulli polynomials having only integer coefficients are finite. We have an infinite chain of subsets: A094960 subset of A366169 subset of A366186 subset of A366187 subset of A366188 subset of ... . See Kellner 2023 (Theorem 13, Conjecture 14, and S_3 (this sequence)).
The sequence is finite and is a supersequence of A366169. It remains to show that 392 is the last term. This is very likely, since the terms depend on the estimate 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. (End)

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, 9 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 (m=1), A366169 (m=2), this sequence (m=3), A366187 (m=4), A366188 (m=5), A366189.

aList := len -> select(n -> denom(diff(bernoulli(n, x), `$`(x, 3))) = 1, [seq(1..len)]): aList(400);

(* From Bernd C. Kellner, Oct 04 2023 (Start) *)
(* k-th derivative of BP *)
k = 3; 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 = 3; Select[Range[1000], DBP[#, k] == 1&]
(* End *)

isok(k) = #select(x->denominator(x)>1, Vec(deriv(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 A366186_gen(): # generator of terms
    return filter(lambda k:k<=3 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x,3)).coeffs()),count(1))
A366186_list = list(islice(A366186_gen(),30)) # Chai Wah Wu, Oct 03 2023

From Bernd C. Kellner, Oct 04 2023: (Start)
Let (n)_m be the falling factorial. Let s_p(n) be the sum of the p-adic digits of n.
The denominator of the third derivative of the n-th Bernoulli polynomial B(n, x) is given as follows (Kellner 2023, Theorem 12).
D_3(n) = 1 for 1 <= n <= 3. For n > 3, D_3(n) = A324370(n-2)/gcd(A324370(n-2), (n)2) = Product{prime p <= (n-1)/(2+((n-1) mod 2)): gcd(p,(n)_3)=1, s_p(n-2) >= p} p.
Then k is a term if and only if D_3(k) = 1. (End)

A366187 Positive integers k such that the fourth 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, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 42, 43, 45, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 66, 67, 68, 70, 71, 72, 73, 78, 79, 80, 81, 91, 92, 93, 110, 111, 121, 122, 123, 143, 147, 156, 157, 171, 176, 177, 178, 190, 191, 192, 210, 211, 255, 392, 393

Offset: 1

Peter Luschny, Oct 03 2023

From Bernd C. Kellner, Oct 04 2023: (Start)
As a published result on Oct 02 2023 (cf. A366169), all such sequences regarding higher derivatives of the Bernoulli polynomials having only integer coefficients are finite. We have an infinite chain of subsets: A094960 subset of A366169 subset of A366186 subset of A366187 subset of A366188 subset of ... . See Kellner 2023 (Theorem 13).
The sequence is finite and is a supersequence of A366186. It remains to show that 393 is the last term. This is very likely, since the terms depend on the estimate 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. (End)

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, 9 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 (m=1), A366169 (m=2), A366186 (m=3), this sequence (m=4), A366188 (m=5), A366189.

aList := len -> select(n -> denom(diff(bernoulli(n, x), `$`(x, 4))) = 1, [seq(1..len)]): aList(400);

(* From Bernd C. Kellner, Oct 04 2023 (Start) *)
(* k-th derivative of BP *)
k = 4; 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 = 4; Select[Range[1000], DBP[#, k] == 1&]
(* End *)

isok(k) = #select(x->denominator(x)>1, Vec(deriv(deriv(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 A366187_gen(): # generator of terms
    return filter(lambda k:k<=4 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x,4)).coeffs()),count(1))
A366187_list = list(islice(A366187_gen(),40)) # Chai Wah Wu, Oct 03 2023

From Bernd C. Kellner, Oct 04 2023: (Start)
Let (n)_m be the falling factorial. Let s_p(n) be the sum of the p-adic digits of n.
The denominator of the fourth derivative of the n-th Bernoulli polynomial B(n, x) is given as follows (Kellner 2023, Theorem 12).
D_4(n) = 1 for 1 <= n <= 4. For n > 4, D_4(n) = A324370(n-3)/gcd(A324370(n-3), (n)3) = Product{prime p <= (n-2)/(2+((n-2) mod 2)): gcd(p,(n)_4)=1, s_p(n-3) >= p} p.
Then k is a term if and only if D_4(k) = 1. (End)

A366188 Positive integers k such that the fifth 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, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 71

Offset: 1

Peter Luschny, Oct 03 2023

From Bernd C. Kellner, Oct 04 2023: (Start)
As a published result on Oct 02 2023 (cf. A366169), all such sequences regarding higher derivatives of the Bernoulli polynomials having only integer coefficients are finite. We have an infinite chain of subsets: A094960 subset of A366169 subset of A366186 subset of A366187 subset of A366188 subset of ... . See Kellner 2023 (Theorem 13).
The sequence is finite and is a supersequence of A366187. It remains to show that 904 is the last term. This is very likely, since the terms depend on the estimate 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. (End)

Peter Luschny, Table of n, a(n) for n = 1..125
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, 9 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 (m=1), A366169 (m=2), A366186 (m=3), A366187 (m=4), (this sequence) (m=5), A366189.

aList := len -> select(n -> denom(diff(bernoulli(n, x), `$`(x, 5))) = 1, [seq(1..len)]): aList(1000);

(* From Bernd C. Kellner, Oct 04 2023 (Start) *)
(* k-th derivative of BP *)
k = 5; 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 = 5; Select[Range[1000], DBP[#, k] == 1&]
(* End *)

isok(k) = #select(x->denominator(x)>1, Vec(deriv(deriv(deriv(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 A366188_gen(): # generator of terms
    return filter(lambda k:k<=5 or all(c.is_integer for c in Poly(diff(bernoulli(k,x),x,5)).coeffs()),count(1))
A366188_list = list(islice(A366188_gen(),30)) # Chai Wah Wu, Oct 03 2023

From Bernd C. Kellner, Oct 04 2023: (Start)
Let (n)_m be the falling factorial. Let s_p(n) be the sum of the p-adic digits of n.
The denominator of the fifth derivative of the n-th Bernoulli polynomial B(n, x) is given as follows (Kellner 2023, Theorem 12).
D_5(n) = 1 for 1 <= n <= 5. For n > 5, D_5(n) = A324370(n-4)/gcd(A324370(n-4), (n)4) = Product{prime p <= (n-3)/(2+((n-3) mod 2)): gcd(p,(n)_5)=1, s_p(n-4) >= p} p.
Then k is a term if and only if D_5(k) = 1. (End)

cp's OEIS Frontend

A324405 Squarefree integers m > 1 such that if prime p divides m, then s_p(m) >= p and s_p(m) == 3 (mod p-1), where s_p(m) is the sum of the base p digits of m.

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A324317 Number of primary Carmichael numbers (A324316) less than 10^n.

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A324318 Number of terms in A324315 (squarefree integers m > 1 such that if prime p divides m, then the sum of the base p digits of m is at least p) less than 10^n.

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

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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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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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