A193267 The number 1 alternating with the numbers A006953/A002445 (which are integers).

1, 2, 1, 4, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 16, 1, 18, 1, 20, 1, 2, 1, 24, 1, 2, 1, 4, 1, 6, 1, 32, 1, 2, 1, 36, 1, 2, 1, 40, 1, 42, 1, 4, 1, 2, 1, 48, 1, 2, 1, 4, 1, 54, 1, 8, 1, 2, 1, 60, 1, 2, 1, 64, 1, 6, 1, 4, 1, 2, 1, 72, 1, 2, 1, 4, 1, 6, 1, 80, 1, 2, 1, 84, 1, 2, 1, 8, 1, 18, 1, 4, 1, 2, 1, 96, 1, 2, 1, 100

Offset: 1

Author

Paul Curtz, Dec 20 2012

Keywords

• nonn

Comments

a(n) is the product over all prime powers p^e, where p^e is the highest power of p dividing n and p-1 divides n. - Peter Luschny, Mar 12 2018

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..2000

Programs

Julia

using Nemo
function A193267(n) P = 1
    for (p, e) in factor(ZZ(n))
        divisible(ZZ(n), p - 1) && (P *= p^e) end
P end
[A193267(n) for n in 1:100] |> println # Peter Luschny, Mar 12 2018

Magma

[Denominator(Bernoulli(n)/n)/Denominator(Bernoulli(n)): n in [1..100]]; // Vincenzo Librandi, Mar 12 2018

Maple

with(numtheory); a := proc(n) divisors(n); map(i->i+1, %); select(isprime, %);
mul(k^padic[ordp](n,k),k=%) end: seq(a(n), n=1..100); # Peter Luschny, Mar 12 2018
# Alternatively:
A193267 := proc(n) local P, F, f, divides; divides := (a,b) -> is(irem(b,a) = 0):
P := 1; F := ifactors(n)[2]; for f in F do if divides(f[1]-1, n) then
P := P*f[1]^f[2] fi od; P end: seq(A193267(n), n=1..100); # Peter Luschny, Mar 12 2018

Mathematica

a[n_] := If[OddQ[n], 1, Denominator[ BernoulliB[n]/n ] / Denominator[ BernoulliB[n]] ]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Dec 21 2012 *)

Formula

a(n+1) = A185633(n+1)/A027760(n+1).

a(n+1) = c(n+2)/c(n+1).