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