A366188 -id:A366188 - cp's OEIS frontend

A324370 Product of all primes p not dividing n such that the sum of the base-p digits of n is at least p, or 1 if no such prime exists.

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, 1430, 2145, 1122, 85, 82110, 2415, 70, 3, 330, 55, 21111090, 285285

Offset: 1

Bernd C. Kellner and Jonathan Sondow, Feb 24 2019

The product is finite, as the sum of the base-p digits of n is n if p > n.
a(198) = 2465 is the only term below 10^6 that is a Carmichael number (A002997).
It appears that a(n)=1 if and only if n is in A094960. - Robert Israel, Mar 30 2020
It turns out that a(n) equals the denominator of the first derivative of the Bernoulli polynomial B(n,x). So a(n)=1 if and only if n is in A094960, also impyling that n+1 is prime. A324370 is also involved in such formulas regarding higher derivatives. See Kellner 2023. - Bernd C. Kellner, Oct 12 2023

			For p = 2, 3, and 5, the sum of the base p digits of 7 is 1+1+1 = 3 >= 2, 2+1 = 3 >= 3, and 1+2 = 3 < 5, respectively, so a(7) = 2*3 = 6.

Robert Israel, Table of n, a(n) for n = 1..5000
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, 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, A094960, A144845, A195441, A324315, A324316, A324317, A324318, A324319, A324320, A324369, A324371, A324404, A324405, A366168, A366169, A366186, A366187, A366188.

N:= 100: # for a(1)..a(N)
V:= Vector(N,1):
p:= 1:
for iter from 1 do
   p:= nextprime(p);
   if p >= N then break fi;
   for n from p+1 to N do
     if n mod p <> 0 and convert(convert(n,base,p),`+`)>= p then
       V[n]:= V[n]*p
     fi
od od:
convert(V,list); # Robert Israel, Mar 30 2020
# Alternatively, note that this formula is suggesting offset 0 and a(0) = 1:
seq(denom(diff(bernoulli(n, x), x)), n = 1..51); # Peter Luschny, Oct 13 2023

SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]];
DD2[n_] := Times @@ Select[Prime[Range[PrimePi[(n+1)/(2+Mod[n+1, 2])]]], !Divisible[n, #] && SD[n, #] >= # &];
Table[DD2[n], {n, 1, 100}]
(* From Bernd C. Kellner, Oct 12 2023 (Start) *)
(* Denominator of first derivative of BP *)
k = 1; Table[Denominator[Together[D[BernoulliB[n, x], {x, k}]]], {n, 1, 100}]
(* End *)

from math import prod
from sympy.ntheory import digits
from sympy import primefactors, primerange
def a(n):
    nonpf = set(primerange(1, n+1)) - set(primefactors(n))
    return prod(p for p in nonpf if sum(digits(n, p)[1:]) >= p)
print([a(n) for n in range(1, 75)]) # Michael S. Branicky, Jul 03 2022

a(n) * A324369(n) = A195441(n-1) = denominator(Bernoulli_n(x) - Bernoulli_n).
a(n) * A324369(n) * A324371(n) = A144845(n-1) = denominator(Bernoulli_{n-1}(x)).
a(n+1) = A195441(n)/A324369(n+1) = A144845(n)/A007947(n+1) = A318256(n). Essentially the same as A318256. - Peter Luschny, Mar 05 2019
From Bernd C. Kellner, Oct 12 2023: (Start)
a(n) = denominator(Bernoulli_n(x)').
k-th derivative: let (n)_m be the falling factorial.
For n > k, a(n-k+1)/gcd(a(n-k+1), (n)_{k-1}) = denominator(Bernoulli_n(x)^(k)). Otherwise, the denominator equals 1. (End)

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)

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.

Original entry on oeis.org

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

Peter Luschny, Oct 03 2023

nonn

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.)

			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.

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

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

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

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);

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 *)

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

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

cp's OEIS Frontend

A324370 Product of all primes p not dividing n such that the sum of the base-p digits of n is at least p, or 1 if no such prime exists.

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A366186 Positive integers k such that the third derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.

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A366187 Positive integers k such that the fourth derivative of the k-th Bernoulli polynomial B(k, x) contains only integer coefficients.

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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.

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