A366168 -id:A366168 - 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)

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

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.

A366426 a(n) = numerator(denominator(Bernoulli''(n, x)) / denominator(Bernoulli(n, 1))).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 3, 5, 5, 7, 1, 5, 15, 1, 1, 7, 7, 1, 1, 1, 1, 77, 7, 35, 3, 1, 1, 455, 35, 7, 21, 55, 55, 7, 7, 7, 105, 1, 5, 221, 13, 11, 33, 55, 1, 19, 1, 5, 15, 1, 1, 5005, 715, 143, 33, 17, 85, 161, 35, 1, 3, 11, 55, 95095

Offset: 0

Peter Luschny, Oct 13 2023

Cf. A366168/A027642, A366427 (denominator), A366570/A366152 (1st derivative).

seq(numer(denom(diff(diff(bernoulli(n, x), x),x))/denom(bernoulli(n, 1))), n = 0..75);

a(n) = numerator(lcm(apply(denominator, Vec(deriv(deriv(bernpol(n))))))/denominator(subst(bernpol(n, x), x, 1))); \\ Michel Marcus, Oct 14 2023

a(n) = numerator(A366168(n) / A027642(n)).

A366427 a(n) = denominator(denominator(Bernoulli''(n, x)) / denominator(Bernoulli(n, 1))).

Original entry on oeis.org

1, 2, 6, 1, 30, 1, 42, 1, 10, 1, 66, 1, 2730, 1, 2, 1, 510, 1, 798, 1, 110, 1, 138, 1, 546, 1, 2, 1, 870, 1, 14322, 1, 170, 1, 6, 1, 1919190, 1, 2, 1, 13530, 1, 1806, 1, 46, 1, 282, 1, 1326, 1, 22, 1, 1590, 1, 798, 1, 290, 1, 354, 1, 56786730, 1, 2, 1, 102, 1

Offset: 0

Peter Luschny, Oct 13 2023

Cf. A366168/A027642, A366426 (numerator), A366570/A366152 (1st derivative).

seq(denom(denom(diff(diff(bernoulli(n, x), x),x))/denom(bernoulli(n, 1))), n = 0..65);

a(n) = denominator(lcm(apply(denominator, Vec(deriv(deriv(bernpol(n))))))/denominator(subst(bernpol(n, x), x, 1))); \\ Michel Marcus, Oct 14 2023

a(n) = denominator(A366168(n) / A027642(n)).

A366572 a(n) = denominator(Bernoulli(n, x)) / denominator(Bernoulli''(n, x)).

Original entry on oeis.org

1, 2, 6, 2, 30, 6, 42, 6, 10, 10, 66, 6, 2730, 210, 2, 6, 510, 10, 798, 42, 110, 330, 138, 2, 546, 546, 2, 2, 870, 30, 14322, 462, 170, 510, 6, 2, 1919190, 51870, 2, 6, 13530, 110, 1806, 42, 46, 690, 1410, 2, 1326, 1326, 22, 66, 1590, 10, 798, 798, 290, 870

Offset: 0

Peter Luschny, Oct 13 2023

nonn

Cf. A144845/A366168, A366571, A144845/A324370 (1st derivative).

seq(denom(bernoulli(n, x))/denom(diff(diff(bernoulli(n, x), x),x)), n=0..100);

a(n) = lcm(apply(denominator, Vec(bernpol(n))))/lcm(apply(denominator, Vec(deriv(deriv(bernpol(n)))))); \\ Michel Marcus, Oct 14 2023

a(n) = A144845(n) / A366168(n).

A366571 a(n) = denominator(Bernoulli(n, x)) / denominator(Bernoulli'(n, x)).

Original entry on oeis.org

1, 2, 6, 1, 30, 1, 42, 1, 10, 1, 66, 1, 2730, 1, 2, 3, 170, 1, 798, 1, 110, 3, 46, 1, 546, 1, 2, 1, 870, 1, 14322, 1, 170, 3, 2, 1, 1919190, 1, 2, 3, 4510, 1, 1806, 1, 46, 15, 94, 1, 1326, 1, 22, 3, 530, 1, 798, 1, 290, 3, 118, 1, 56786730, 1, 2, 3, 34, 5, 64722

Offset: 0

Peter Luschny, Oct 13 2023

Cf. A144845/A324370, A366572, A144845/A366168 (2nd derivative).

seq(denom(bernoulli(n, x))/denom(diff(bernoulli(n, x), x)), n = 0..66);

a(n) = lcm(apply(denominator, Vec(bernpol(n))))/lcm(apply(denominator, Vec(deriv(bernpol(n))))); \\ Michel Marcus, Oct 14 2023

a(n) = A144845(n) / A324370(n).

A366573 a(n) = denominator(Bernoulli'(n, x)) / denominator(Bernoulli''(n, x)).

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 6, 1, 10, 1, 6, 1, 210, 1, 2, 3, 10, 1, 42, 1, 110, 3, 2, 1, 546, 1, 2, 1, 30, 1, 462, 1, 170, 3, 2, 1, 51870, 1, 2, 3, 110, 1, 42, 1, 46, 15, 2, 1, 1326, 1, 22, 3, 10, 1, 798, 1, 290, 3, 2, 1, 930930, 1, 2, 3, 34, 5, 966, 1, 2, 3, 110, 1

Offset: 0

Peter Luschny, Oct 13 2023

nonn

Cf. A324370/A366168, A366572, A144845/A366168.

seq(denom(diff(bernoulli(n, x), x))/denom(diff(diff(bernoulli(n, x), x),x)), n = 0..100);

a(n) = lcm(apply(denominator, Vec(deriv(bernpol(n)))))/ lcm(apply(denominator, Vec(deriv(deriv(bernpol(n)))))); \\ Michel Marcus, Oct 14 2023

a(n) = A324370(n) / A366168(n).

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A366426 a(n) = numerator(denominator(Bernoulli''(n, x)) / denominator(Bernoulli(n, 1))).

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

Formula

A366427 a(n) = denominator(denominator(Bernoulli''(n, x)) / denominator(Bernoulli(n, 1))).

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

Formula

A366572 a(n) = denominator(Bernoulli(n, x)) / denominator(Bernoulli''(n, x)).

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

Formula

A366571 a(n) = denominator(Bernoulli(n, x)) / denominator(Bernoulli'(n, x)).

Views

Author

Keywords

Crossrefs

Programs

Maple

PARI

Formula

A366573 a(n) = denominator(Bernoulli'(n, x)) / denominator(Bernoulli''(n, x)).

Views