A195441 - OEIS

%I #95 Sep 10 2024 08:15:55

%S 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,

%T 14,2,30,2,462,231,3570,210,6,2,51870,2730,210,42,2310,330,2310,210,

%U 4830,210,210,210,6630,1326,858,66,330,110,798,114,870,30,30,6

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

%C 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).

%C a(n) is squarefree, by the von Staudt-Clausen theorem on the denominators of Bernoulli numbers. - _Kieren MacMillan_ and _Jonathan Sondow_, Nov 20 2015

%C 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

%C 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

%C 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

%H Peter Luschny, <a href="/A195441/b195441.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)

%H Olivier Bordellès, Florian Luca, Pieter Moree, and Igor E. Shparlinski, <a href="https://doi.org/10.1112/S0025579318000153">Denominators of Bernoulli polynomials</a>, Mathematika 64 (2018), 519-541.

%H Harald Hofstätter, <a href="https://arxiv.org/abs/2010.03440">Denominators of coefficients of the Baker-Campbell-Hausdorff series</a>, arXiv:2010.03440 [math.NT], 2020. Mentions this sequence.

%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.

%F a(n) = A064538(n)/(n+1). - _Jonathan Sondow_, Nov 12 2015

%F A001221(a(n)) = A001222(a(n)). - _Kieren MacMillan_ and _Jonathan Sondow_, Nov 20 2015

%F a(2*n)/a(2*n+1) = A286516(n+1). - _Bernd C. Kellner_ and _Jonathan Sondow_, May 24 2017

%F a(n) = A007947(A338025(n+1)). - _Harald Hofstätter_, Oct 10 2020

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

%F Note that the formulas here are shifted in index by 1 due to the definition of a(n) using index n+1!

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

%F a(n) = A144845(n) / A324371(n+1).

%F a(n-1) = lcm(a(n), rad(n+1)), if n >= 3 is odd.

%F If n+1 is composite, then rad(n+1) divides a(n-1).

%F If m is a Carmichael number (A002997), then m divides both a(m-1) and a(m-2).

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

%p A195441 := n -> denom(bernoulli(n+1, x)-bernoulli(n+1)):

%p seq(A195441(i),i=0..59);

%p # Formula of Kellner and Sondow:

%p a := proc(n) local s; s := (p,n) -> add(i,i=convert(n,base,p));

%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

%t a[n_] := Denominator[Together[(BernoulliB[n + 1, x] - BernoulliB[n + 1])]]; Table[a[n], {n, 0, 59}] (* _Jonathan Sondow_, Nov 20 2015 *)

%t 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 *)

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

%o (Sage)

%o 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])

%o print([A195441(n) for n in (0..59)]) # _Peter Luschny_, May 14 2017

%o (Julia)

%o using Nemo, Primes

%o function A195441(n::Int)

%o     n < 4 && return ZZ([1,1,2,1][n+1])

%o     P = primes(2, div(n+2, 2+n%2))

%o     prod([ZZ(p) for p in P if p <= sum(digits(n+1, base=p))])

%o end

%o println([A195441(n) for n in 0:59]) # _Peter Luschny_, May 14 2017

%o (Python)

%o from math import prod

%o from sympy.ntheory.factor_ import primerange, digits

%o 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

%Y Cf. A002997, A064538, A094960, A144845, A286516, A286762, A286763, A318256, A324369, A324370, A324371.

%K nonn

%O 0,3

%A _Peter Luschny_, Sep 18 2011

%E Definition simplified by _Jonathan Sondow_, Nov 20 2015