A144845 - OEIS

%I #96 Jun 05 2024 10:02:24

%S 1,2,6,2,30,6,42,6,30,10,66,6,2730,210,30,6,510,30,3990,210,2310,330,

%T 690,30,2730,546,42,14,870,30,14322,462,39270,3570,210,6,1919190,

%U 51870,2730,210,94710,2310,99330,2310,4830,4830,9870,210,46410,6630,14586,858

%N Least number k such that all coefficients of k*B(n,x), the n-th Bernoulli polynomial, are integers.

%C 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.

%C This sequence appears in a paper of Bazsó & Mező, who use this sequence to give necessary and sufficient conditions for power sums to be integer polynomials. - _Istvan Mezo_, Mar 20 2016

%C 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

%H Bernd C. Kellner, <a href="/A144845/b144845.txt">Table of n, a(n) for n = 0..10000</a> (n = 0..1000 from T. D. Noe)

%H András Bazsó and István Mező, <a href="http://dx.doi.org/10.1016/j.jnt.2015.01.019">On the coefficients of power sums of arithmetic progressions</a>, J. Number Th., 153 (2015), 117-123.

%H András Bazsó and István Mező, <a href="https://www.emis.de/journals/JIS/VOL21/Mezo/mezo25.html">Some Notes on Alternating Power Sums of Arithmetic Progressions</a>, J. Int. Seq., Vol. 21 (2018), Article 18.7.8.

%H Bernd C. Kellner, <a href="https://doi.org/10.1016/j.jnt.2017.03.020">On a product of certain primes</a>, J. Number Theory, 179 (2017), 126-141; arXiv:<a href="https://arxiv.org/abs/1705.04303">1705.04303</a> [math.NT], 2017.

%H Bernd C. Kellner, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL27/Kellner/kell2.html">On the finiteness of Bernoulli polynomials whose derivative has only integral coefficients</a>, J. Integer Seq. 27 (2024), Article 24.2.8, 11 pp.; arXiv:<a href="https://arxiv.org/abs/2310.01325">2310.01325</a> [math.NT], 2023.

%H Bernd C. Kellner and Jonathan Sondow, <a href="https://doi.org/10.4169/amer.math.monthly.124.8.695">Power-Sum Denominators</a>, Amer. Math. Monthly, 124 (2017), 695-709; arXiv:<a href="https://arxiv.org/abs/1705.03857">1705.03857</a> [math.NT], 2017.

%H Bernd C. Kellner and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/s95/s95.pdf">The denominators of power sums of arithmetic progressions</a>, Integers 18 (2018), #A95, 17 pp.; arXiv:<a href="https://arxiv.org/abs/1705.05331">1705.05331</a> [math.NT], 2017.

%H Bernd C. Kellner and Jonathan Sondow, <a href="http://math.colgate.edu/~integers/v52/v52.pdf">On Carmichael and polygonal numbers, Bernoulli polynomials, and sums of base-p digits</a>, Integers 21 (2021), #A52, 21 pp.; arXiv:<a href="https://arxiv.org/abs/1902.10672">1902.10672</a> [math.NT], 2019.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BernoulliPolynomial.html">Bernoulli Polynomial</a>.

%F From _Bernd C. Kellner_, Oct 18 2023: (Start)

%F 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).

%F a(n) = lcm(A195441(n-1), A027642(n)) = lcm(denom(B(n,x)-B_n), denom(B_n)) = denom(B(n,x)).

%F a(n) = lcm(A195441(n), rad(n+1)).

%F a(n) = lcm(a(n+1), rad(n+1)), if n >= 2 is even.

%F a(2n)/a(2n+1) = A286517(n), if n >= 1.

%F a(n) = A324369(n+1) * A324370(n+1) * A324371(n+1).

%F a(n) = A324370(n+1) * rad(n+1).

%F a(n) = rad(A064538(n)).

%F 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}).

%F (See papers of Kellner and Kellner & Sondow.) (End)

%p seq(denom(bernoulli(i,x)),i=0..51); # _Peter Luschny_, Jun 16 2012

%t (* From _Bernd C. Kellner_, Oct 18 2023: (Start) *)

%t (* Denominator formula *)

%t Table[Denominator[Together[BernoulliB[n, x]]], {n, 0, 51}]

%t (* Product formula *)

%t 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, #] >= # &];

%t DB[n_] := DD2[n+1] rad[n+1]; Table[DB[n], {n, 0, 51}]

%t (* (End) *)

%o (Sage)

%o def A144845(n):

%o     return mul(prime_divisors(n+1) + [p for p in (2..(n+2)//(2+n%2))

%o     if is_prime(p) and not p.divides(n+1) and sum((n+1).digits(base=p)) >= p])

%o print([A144845(n) for n in (0..51)]) # _Peter Luschny_, Sep 12 2018

%o (PARI) a(n) = lcm(apply(x->denominator(x), Vec(bernpol(n)))); \\ _Michel Marcus_, Mar 03 2020

%Y Cf. A027642, A053383, A064538, A195441, A286515, A286516, A286517, A324369, A324370, A324371.

%K nonn

%O 0,2

%A _T. D. Noe_, Sep 22 2008