A286762 -id:A286762 - cp's OEIS frontend

A195441 a(n) = denominator(Bernoulli_{n+1}(x) - Bernoulli_{n+1}).

1, 1, 2, 1, 6, 2, 6, 3, 10, 2, 6, 2, 210, 30, 6, 3, 30, 10, 210, 42, 330, 30, 30, 30, 546, 42, 14, 2, 30, 2, 462, 231, 3570, 210, 6, 2, 51870, 2730, 210, 42, 2310, 330, 2310, 210, 4830, 210, 210, 210, 6630, 1326, 858, 66, 330, 110, 798, 114, 870, 30, 30, 6

Offset: 0

Peter Luschny, Sep 18 2011

nonn

If s(n) is the smallest number such that s(n)*(1^n + 2^n + ... + x^n) is a polynomial in x with integer coefficients then a(n)=s(n)/(n+1) (see A064538).
a(n) is squarefree, by the von Staudt-Clausen theorem on the denominators of Bernoulli numbers. - Kieren MacMillan and Jonathan Sondow, Nov 20 2015
Kellner and Sondow give a detailed analysis of this sequence and provide a simple way to compute the terms without using Bernoulli polynomials and numbers. They prove that a(n) is the product of the primes less than or equal to (n+2)/(2+(n mod 2)) such that the sum of digits of n+1 in base p is at least p. - Peter Luschny, May 14 2017
The equation a(n-1) = denominator(Bernoulli_n(x) - Bernoulli_n) = rad(n+1) has only finitely many solutions, where rad(n) = A007947(n) is the radical of n. It is conjectured that S = {3, 5, 8, 9, 11, 27, 29, 35, 59} is the full set of all such solutions. Note that (S\{8})+1 joined with {1,2} equals A094960. More precisely, the set S implies the finite sequence of A094960. See Kellner 2023. - Bernd C. Kellner, Oct 18 2023
As was observed in the example section of A318256: denominator(B_n(x)) = rad(n+1) if n is in {0, 1, 3, 5, 9, 11, 27, 29, 35, 59} = {A094960(n) - 1: 1 <= n <= 10}. - Peter Luschny, Oct 18 2023

Peter Luschny, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Olivier Bordellès, Florian Luca, Pieter Moree, and Igor E. Shparlinski, Denominators of Bernoulli polynomials, Mathematika 64 (2018), 519-541.
Harald Hofstätter, Denominators of coefficients of the Baker-Campbell-Hausdorff series, arXiv:2010.03440 [math.NT], 2020. Mentions this sequence.
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. A002997, A064538, A094960, A144845, A286516, A286762, A286763, A318256, A324369, A324370, A324371.

using Nemo, Primes
function A195441(n::Int)
    n < 4 && return ZZ([1,1,2,1][n+1])
    P = primes(2, div(n+2, 2+n%2))
    prod([ZZ(p) for p in P if p <= sum(digits(n+1, base=p))])
end
println([A195441(n) for n in 0:59]) # Peter Luschny, May 14 2017

A195441 := n -> denom(bernoulli(n+1, x)-bernoulli(n+1)):
seq(A195441(i),i=0..59);
# Formula of Kellner and Sondow:
a := proc(n) local s; s := (p,n) -> add(i,i=convert(n,base,p));
select(isprime,[$2..(n+2)/(2+irem(n,2))]); mul(i,i=select(p->s(p,n+1)>=p,%)) end: seq(a(n), n=0..59); # Peter Luschny, May 14 2017

a[n_] := Denominator[Together[(BernoulliB[n + 1, x] - BernoulliB[n + 1])]]; Table[a[n], {n, 0, 59}] (* Jonathan Sondow, Nov 20 2015 *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus @@ IntegerDigits[n, p]]; DD[n_] := Times @@ Select[Prime[Range[PrimePi[(n+2)/(2+Mod[n, 2])]]], SD[n+1, #] >= # &]; Table[DD[n], {n, 0, 59}] (* Bernd C. Kellner, Oct 18 2023 *)

a(n) = {my(vp = Vec(bernpol(n+1, x)-bernfrac(n+1))); lcm(vector(#vp, k, denominator(vp[k])));} \\ Michel Marcus, Feb 08 2016

from math import prod
from sympy.ntheory.factor_ import primerange, digits
def A195441(n): return prod(p for p in primerange((n+2)//(2|n&1)+1) if sum(digits(n+1,p)[1:])>=p) # Chai Wah Wu, Oct 04 2023

A195441 = lambda n: mul([p for p in (2..(n+2)//(2+n%2)) if is_prime(p) and sum((n+1).digits(base=p))>=p])
print([A195441(n) for n in (0..59)]) # Peter Luschny, May 14 2017

a(n) = A064538(n)/(n+1). - Jonathan Sondow, Nov 12 2015
A001221(a(n)) = A001222(a(n)). - Kieren MacMillan and Jonathan Sondow, Nov 20 2015
a(2*n)/a(2*n+1) = A286516(n+1). - Bernd C. Kellner and Jonathan Sondow, May 24 2017
a(n) = A007947(A338025(n+1)). - Harald Hofstätter, Oct 10 2020
From Bernd C. Kellner, Oct 18 2023: (Start)
Note that the formulas here are shifted in index by 1 due to the definition of a(n) using index n+1!
a(n) = A324369(n+1) * A324370(n+1).
a(n) = A144845(n) / A324371(n+1).
a(n-1) = lcm(a(n), rad(n+1)), if n >= 3 is odd.
If n+1 is composite, then rad(n+1) divides a(n-1).
If m is a Carmichael number (A002997), then m divides both a(m-1) and a(m-2).
See papers of Kellner and Kellner & Sondow. (End)

Definition simplified by Jonathan Sondow, Nov 20 2015

A064538 a(n) is the smallest positive integer such that a(n)*(1^n + 2^n + ... + x^n) is a polynomial in x with integer coefficients.

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 42, 24, 90, 20, 66, 24, 2730, 420, 90, 48, 510, 180, 3990, 840, 6930, 660, 690, 720, 13650, 1092, 378, 56, 870, 60, 14322, 7392, 117810, 7140, 210, 72, 1919190, 103740, 8190, 1680, 94710, 13860, 99330, 9240, 217350, 9660, 9870, 10080, 324870

Offset: 0

Floor van Lamoen, Oct 08 2001

a(n) is a multiple of n+1. - Vladimir Shevelev, Dec 20 2011
Let P_n(x) = 1^n + 2^n + ... + x^n = Sum_{i=1..n+1}c_i*x^i. Let P^*n(x) = Sum{i=1..n+1}(c_i/(i+1))*(x^(i+1)-x). Then b(n) = (n+1)*a(n+1)is the smallest positive integer such that b(n)*P^*n(x) is a polynomial with integer coefficients. Proof follows from the recursion P(n+1)(x) = x + (n+1)*P^*n(x). As a corollary, note that, if p is the maximal prime divisor of a(n), then p<=n+1. - _Vladimir Shevelev, Dec 21 2011
The recursion P_(n+1)(x) = x + (n+1)*P^*n(x) is due to Abramovich (1973); see also Shevelev (2007). - _Jonathan Sondow, Nov 16 2015
The sum S_m(n) = Sum_{k=0..n} k^m can be written as S_m(n) = n(n+1)(2n+1)P_m(n)/a(m) for even m>1, or S_m(n) = n^2*(n+1)^2*P_m(n)/a(m) for odd m>1, where a(m) is the LCM of the denominators of the coefficients of the polynomial P_m/a(m), i.e., the smallest integer such that P_m defined in this way has integer coefficients. (Cf. Michon link.) - M. F. Hasler, Mar 10 2013
a(n)/(n+1) is squarefree, by Faulhaber's formula and the von Staudt-Clausen theorem on the denominators of Bernoulli numbers. - Kieren MacMillan and Jonathan Sondow, Nov 20 2015
a(n) equals n+1 times the product of the primes p <= (n+2)/(2+(n mod 2)) such that the sum of the base-p digits of n+1 is at least p. - Bernd C. Kellner and Jonathan Sondow, May 24 2017

			1^3 + 2^3 + ... + x^3 = (x(x+1))^2/4 so a(3)=4.
1^4 + 2^4 + ... + x^4 = x(x+1)(2x+1)(3x^2+3x-1)/30, therefore a(4)=30.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprints), p. 804, Eq. 23.1.4.

Chai Wah Wu, Table of n, a(n) for n = 0..10000 (n = 0..1000 from T. D. Noe)
V. S. Abramovich, Power sums of natural numbers, Kvant 5 (1973), 22-25. (in Russian)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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.
Meng Fai Lim, On the p-divisibility of even K-groups of the ring of integers of a cyclotomic field, arXiv:2308.04099 [math.NT], 2023.
Dr. Math, Summing n^k.
R. Mestrovic, A congruence modulo n^3 involving two consecutive sums of powers and its applications, arXiv:1211.4570 [math.NT], 2012.
R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
G. Michon, Faulhaber's Formula on NUMERICANA.com.
E. S. Rowland, Sums of Consecutive Powers
Vladimir Shevelev, A Short Proof of a Known Relation for Consecutive Power Sums, arXiv:0711.3692 [math.CA], 2007.
Eric Weisstein's World of Mathematics, Power Sum
Wikipedia, Faulhaber's Formula.

Cf. A144845, A195441, A256581, A286516, A286762, A286763, A324369, A324370, A324371.

A064538 := n -> denom((bernoulli(n+1,x)-bernoulli(n+1))/(n+1)): # Peter Luschny, Aug 19 2011
# Formula of Kellner and Sondow (2017):
a := proc(n) local s; s := (p,n) -> add(i,i=convert(n,base,p));
select(isprime,[$2..(n+2)/(2+irem(n,2))]);
(n+1)*mul(i,i=select(p->s(p,n+1)>=p,%)) end: seq(a(n), n=0..48); # Peter Luschny, May 14 2017

A064538[n_] := Denominator[ Together[ (BernoulliB[n+1, x] - BernoulliB[n+1])/(n+1)]];
Table[A064538[n], {n, 0, 44}] (* Jean-François Alcover, Feb 21 2012, after Maple *)

a(n) = {my(vp = Vec(bernpol(n+1, x)-bernfrac(n+1))/(n+1)); lcm(vector(#vp, k, denominator(vp[k])));} \\ Michel Marcus, Feb 07 2016

from _future_ import division
from sympy.ntheory.factor_ import digits, nextprime
def A064538(n):
    p, m = 2, n+1
    while p <= (n+2)//(2+ (n% 2)):
        if sum(d for d in digits(n+1,p)[1:]) >= p:
            m *= p
        p = nextprime(p)
    return m # Chai Wah Wu, Mar 07 2018

A064538 = lambda n: (n+1)*mul([p for p in (2..(n+2)//(2+n%2)) if is_prime(p) and sum((n+1).digits(base=p)) >= p])
print([A064538(n) for n in (0..48)]) # Peter Luschny, May 14 2017

a(n) = (n+1)*A195441(n). - Jonathan Sondow, Nov 12 2015
A001221(a(n)/(n+1)) = A001222(a(n)/(n+1)). - Kieren MacMillan and Jonathan Sondow, Nov 20 2015
rad(a(n)) = A007947(a(n)) = A144845(n) = A324369(n+1) * A324370(n+1) * A324371(n+1). - Bernd C. Kellner, Oct 12 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).

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

A286763 Numbers that appear in A195441 at least once for two consecutive indices.

Original entry on oeis.org

1, 30, 210, 330, 2310, 3990, 6090, 14790, 43890, 66990, 82110, 125970, 144210, 181830, 881790, 1009470, 1067430, 1217370, 2284590, 2381190, 17687670, 18888870, 26265030, 35068110, 39544890, 47763870, 115223790, 127652070, 406816410, 497668710, 741110370, 1024748670

Offset: 1

Peter Luschny, May 14 2017

nonn

The sequence is infinite; see Cor. 3 in "The denominators of power sums of arithmetic progressions". - Bernd C. Kellner and Jonathan Sondow, May 24 2017

			A195441(21) = A195441(22) = 30, so 30 is in the sequence. - _Jonathan Sondow_, Dec 11 2018

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), Article #A95, 17 pp.; arXiv:1705.05331 [math.NT], 2017.

Cf. A195441, A286516, A286762.

function A286763_search()
    A = fmpz[]; a = fmpz(0)
    for n in 0:10000
        u = A195441(n)
        a == u && push!(A, a)
        a = u
    end
    S = sort([a for a in Set(A)])
S[1:32] end
println(A286763_search())

Take[#, 32] &@ Union@ SequenceCases[ Table[ Denominator[ Together[ (BernoulliB[n + 1, x] - BernoulliB[n + 1])]], {n, 0, 2000}], w_ /; And[SameQ @@ w, Length@ w >= 2]][[All, 1]] (* Michael De Vlieger, Sep 22 2017, after Jonathan Sondow at A195441 *)

cp's OEIS Frontend

A195441 a(n) = denominator(Bernoulli_{n+1}(x) - Bernoulli_{n+1}).

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A064538 a(n) is the smallest positive integer such that a(n)*(1^n + 2^n + ... + x^n) is a polynomial in x with 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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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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A286763 Numbers that appear in A195441 at least once for two consecutive indices.

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Mathematica

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