A366189 a(n) is the positive integer k such that the k-th derivative of the n-th Bernoulli polynomial B(n, x) contains only integer coefficients but no lower derivative of B(n, x) has this property.

1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 3, 2, 2, 3, 3, 4, 3, 2, 3, 4, 5, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 3, 1, 2, 3, 4, 5, 6, 3, 4, 5, 4, 5, 6, 7, 5, 3, 4, 3, 4, 5, 2, 3, 2, 3, 4, 1, 2, 3, 4, 5, 6, 3, 4, 4, 5, 2, 3, 3, 4, 5, 6, 7, 6, 3, 4, 3, 4, 5, 6, 5, 6, 7

Offset: 1

Author

Peter Luschny, Oct 03 2023

Keywords

• nonn

Comments

The 'integral index' k of a rational polynomial p(x) is the smallest integer k such that p^[k](x) is an integer polynomial, where p^[k](x) denotes the k-th derivative of p. (Integer polynomials have integral index 0.) Using this way of speaking, the a(n) are the integral indices of the Bernoulli polynomials.

Conjecture: Every integer appears in this sequence only a finite number of times. (This generalizes the conjectures made in A366186-A366188.)

Examples

			B = Bernoulli(8, x).
B = -(1/30) + (2/3)*x^2 - (7/3)*x^4 + (14/3)*x^6 - 4*x^7 + x^8;
B' = (4/3)*x - (28/3)*x^3 + 28*x^5 - 28*x^6 + 8*x^7;
B'' = (4/3) - 28*x^2 + 140*x^4 - 168*x^5 + 56*x^6;
B''' = -56*x + 560*x^3 - 840*x^4 + 336*x^5.
Thus the integral index of B is a(8) = 3.
		

Links

• Peter Luschny, Table of n, a(n) for n = 1..4000

Crossrefs

Bernoulli polynomials: A196838/A196839 (with rising powers).

Cf. A094960 (m=1), A366169 (m=2), A366186 (m=3), A366187 (m=4), A366188 (m=5).

Programs

Maple

aList := proc(len) local n, k, d, A;
    A := Array([seq(0, n = 0..len-1)]);
    for n from 1 to len do
       k := 0: d:= 0;
       while d <> 1 do
          k := k + 1;
          d := denom(diff(bernoulli(n, x), `$`(x, k)));
       od;
       A[n] := k;
    od;
convert(A, list) end:
aList(86);

Mathematica

a[n_] := Module[{b, k, x}, b = BernoulliB[n, x]; For[k = 1, True, k++, b = D[b, x]; If[AllTrue[CoefficientList[b, x], IntegerQ], Return[k]]]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 23 2023 *)

Python

from itertools import count
from sympy import Poly, bernoulli, diff
from sympy.abc import x
def A366189(n):
    p = Poly(bernoulli(n,x))
    for i in count(1):
        p = diff(p)
        if all(c.is_integer for c in p.coeffs()):
            return i # Chai Wah Wu, Oct 03 2023

SageMath

def A366189List(len):
    A = [0 for _ in range(len)]
    P. = ZZ[]
    for n in range(len):
        ber = bernoulli_polynomial(x, n + 1)
        k = 0
        while True:
            k = k + 1
            ber = diff(ber, x)
            if ber.denominator() == 1:
                A[n] = k; break
    return A
print(A366189List(86))  # Peter Luschny, Oct 04 2023