login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A144845 Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers. 28
1, 2, 6, 2, 30, 6, 42, 6, 30, 10, 66, 6, 2730, 210, 30, 6, 510, 30, 3990, 210, 2310, 330, 690, 30, 2730, 546, 42, 14, 870, 30, 14322, 462, 39270, 3570, 210, 6, 1919190, 51870, 2730, 210, 94710, 2310, 99330, 2310, 4830, 4830, 9870, 210, 46410, 6630, 14586, 858 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
COMMENTS
The lcm of the terms in row n of A053383. It appears that the Bernoulli polynomial B(n,x) is irreducible for all even n.
This sequence appears in a paper of Bazsó & Mező who use this sequence to give necessary and sufficient condition for power sums to be integer polynomials. - Istvan Mezo, Mar 20 2016
In "The denominators of power sums of arithmetic progressions" Corollary 1, we give a simple way to compute a(n) without using Bernoulli polynomials. Namely, a(n) equals (product of the primes dividing n+1) times (product of the primes p <= (n+1)/(2+(n+1 mod 2)) not dividing n+1 such that the sum of the base-p digits of n+1 is at least p). - Bernd C. Kellner and Jonathan Sondow, May 15 2017
LINKS
Bernd C. Kellner, Table of n, a(n) for n = 0..10000 (n = 0..1000 from T. D. Noe)
András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.
András Bazsó and István Mező, Some Notes on Alternating Power Sums of Arithmetic Progressions, J. Int. Seq., Vol. 21 (2018), Article 18.7.8.
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.
Eric Weisstein's World of Mathematics, Bernoulli Polynomial
FORMULA
From Bernd C. Kellner, Oct 18 2023: (Start)
Let rad(n) = A007947(n) be the radical of n. Let (n)_m be the falling factorial. Let f^(m)(x) denote the m-th derivative of f(x).
a(n) = lcm(A195441(n-1), A027642(n)) = lcm(denom(B(n,x)-B_n), denom(B_n)) = denom(B(n,x)).
a(n) = lcm(A195441(n), rad(n+1)).
a(n) = lcm(a(n+1), rad(n+1)), if n >= 2 is even.
a(2n)/a(2n+1) = A286517(n), if n >= 1.
a(n) = A324369(n+1) * A324370(n+1) * A324371(n+1).
a(n) = A324370(n+1) * rad(n+1).
a(n) = rad(A064538(n)).
If n >= m >= 1, then denom(B^(m)(n,x)) = a(n-m)/gcd(a(n-m), (n)_m) = A324370(n-m+1)/gcd(A324370(n-m+1), (n)_{m-1}).
(See papers of Kellner and Kellner & Sondow.) (End)
MAPLE
seq(denom(bernoulli(i, x)), i=0..51); # Peter Luschny, Jun 16 2012
MATHEMATICA
(* From Bernd C. Kellner, Oct 18 2023: (Start) *)
(* Denominator formula *)
Table[Denominator[Together[BernoulliB[n, x]]], {n, 0, 51}]
(* Product formula *)
SD[n_, p_] := If[n < 1 || p < 2, 0, Plus@@IntegerDigits[n, p]]; rad[n_] := Times @@ Select[Divisors[n], PrimeQ]; (* A324370 *) DD2[n_] := Times @@ Select[Prime[Range[PrimePi[(n+1)/(2+Mod[n+1, 2])]]], !Divisible[n, #] && SD[n, #] >= # &];
DB[n_] := DD2[n+1] rad[n+1]; Table[DB[n], {n, 0, 51}]
(* (End) *)
PROG
(Sage)
def A144845(n):
return mul(prime_divisors(n+1) + [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([A144845(n) for n in (0..51)]) # Peter Luschny, Sep 12 2018
(PARI) a(n) = lcm(apply(x->denominator(x), Vec(bernpol(n)))); \\ Michel Marcus, Mar 03 2020
CROSSREFS
Sequence in context: A284004 A225481 A366572 * A346093 A345284 A200563
KEYWORD
nonn
AUTHOR
T. D. Noe, Sep 22 2008
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 19 06:32 EDT 2024. Contains 370953 sequences. (Running on oeis4.)