# 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: