Generalized Clausen numbers:
definition and application.

Abstract. Generalized Clausen numbers are introduced and the relation to the Bernoulli numbers of order n discussed. (Clausen, Thomas, "Lehrsatz aus einer Abhandlung über die Bernoullischen Zahlen", Astr. Nachr. 17 (1840), 351-352.)

Let Cn denote the Clausen numbers. They are defined as C(n) = Product_{ p - 1 | n} p, where p is prime.

The generalized Clausen numbers Cn,k are defined as C(n, k) = Product_{ p - k | n} p, where p is prime.

The special case k=0, C(n,0) is the square-free kernel of n (A007947 at OEIS). The classical Clausen numbers C(n) = C(n,1) are listed in sequence A141056 at OEIS.
The table below is A160014

k\n 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 1 2 3 2 5 6 7 2 3 10 11 6
1 1 2 6 2 30 2 42 2 30 2 66 2 2730
2 1 3 3 15 3 21 15 3 3 165 21 39 15
3 1 1 5 1 35 1 5 1 385 1 65 1 35
4 1 5 5 35 5 5 35 55 5 455 5 5 35
5 1 1 7 1 7 1 77 1 91 1 7 1 1309


The distinct Clausen numbers.

For fixed k let C'(n, k) = { Product_{ p - k | n} p (prime); n = 0,1,2..}

read as a set (distinct elements), sorted in the natural way. C'(n, 0) are the square-free numbers A005117 and C'(n, 1) are the distinct classical Clausen numbers A090801 .
 

k\n 0 1 2 3 4 5 6 7 8 9 10 11 12
0 1 2 3 5 6 7 10 11 13 14 15 17 19
1 1 2 6 30 42 66 138 282 330 354 498 510 642
2 1 3 15 21 39 57 93 129 165 183 195 219 273
3 1 5 35 65 85 145 185 205 305 385 445 485 545
4 1 5 35 55 85 115 145 205 235 295 355 385 415
5 1 7 77 91 133 217 301 469 511 553 679 889 973


How to compute the Clausen numbers.

The Clausen numbers are the denominator of the Bernoulli numbers. They can be computed as explained by Clausen:

Der Bruch der n-ten Bernoullischen Zahl wird so gefunden: Man addire zu den Theilern von 2n ... 1, 2, a, a', a", ..., 2n die Einheit, wodurch man die Reihe Zahlen 2, 3, a + 1, a' + 1, ..., 2n+1 bekommt. Aus dieser nimmt man bloß die Primzahlen 2, 3, p, p' etc. und bildet den Bruch der n-ten Bernoullischen Zahl ... (as cited by R. Fritsch.)

An almost verbatim translation to a notation similar to Maple gives:

Clausen := proc(n, k) local S;
S := divisors( n );
S := map(i -> i + k, S);
S := select(isprime, S);
product( S ) end:

The distinct Clausen numbers can be computed (Maple) as:

DistinctClausen := proc(n, k) local Clausen;
Clausen := proc(n, k) local i;
mul(i, i = select(isprime,
map(i -> i + k, numtheory[divisors](n)))) end:
sort(convert(remove(m -> m > n,
{seq(Clausen(j, k), j = 0..10*n )}), list)) end:

The complement of the (distinct) Clausen numbers.

{1, 2, 3, 4, ...} \ C'(n,0) = { n | Möbius(n) = 0} = A013929.

Application: Bernoulli numbers.

The denominators of the Bernoulli numbers Bn are displayed in A027642 as:

1, 2, 6, 1, 30, 1, 42, 1, 30, 1, 66, 1, 2730, 1, 6, 1, 510, 1, ..

This sequence of denominators of Bn is defined by convention, not by necessity. The convention amounts to map 0 to the rational number 0/1. It might be more appropriate to regard numerators and denominators of the Bernoulli numbers as independent sequences Nn and Dn which combine to Bn = Nn/Dn. This is suggested by the theorem of Clausen which describes the denominators as the sequence Dn = 1, 2, 6, 2, 30, 2, 42,... which combines with Nn = 1, -1, 1, 0, -1, 0,... to the sequence of Bernoulli numbers. (Cf. A141056 and A027760.)

More general: Let B(k)n(x) denote the Bernoulli polynomials of order k, defined by the generating function

(t/(exp(t)-1))^k*exp(x*t) = Sum_{n>=0} B_n{^(k)}(x) t^n/n!

Bernoulli numbers of order 1 (defined as B(1)n(1)) can be regarded as a pair of sequences B(1)n = N(1)n/D(1)n with N(1)n = A027641, D(1)n = C(n).

Similarly Bernoulli numbers of order 2 (defined as B(2)n(1)) can be regarded as a pair of sequences B(2)n = N(2)n/D(2)n with D(2)n =  C(n).  N(2)n are the values of the following polynomials at x = 1.

The Clausen-normalized Bernoulli polynomials of order 2.

1
2 x - 2
6 x^2 - 12 x + 5
2 x^3 - 6 x^2 + 5 x - 1
30 x^4 - 120 x^3 + 150 x^2 - 60 x + 3
2 x^5 - 10 x^4 + 50/3 x^3 - 10 x^2 + x + 1/3
42 x^6 - 252x^5 + 525x^4 - 420x^3 + 63x^2 + 42x - 5

The sequence N(2)n = Numerator(B(2)n) is A160035 at OEIS and starts

1, 0, -1, 0, 3, 0, -5, 0, 7, 0, -45, 0, 7601, 0, -91, 0, 54255,..

These numbers can be computed as (Maple)

a := proc(n) local g, c, i;
g := k -> (t / (exp(t) - 1))^k*exp(x*t):
c := proc(n) local i;
mul(i, i=select(isPrime, map(i->i+1,divisors(n)))) end:
convert(series(g(2), t, n + 8), polynom):
seq(i!*c(i)*subs(x = 1, coeff(%, t, i)), i = 0..n) end: