A160014 Generalized Clausen numbers (table read by antidiagonals).

1, 1, 1, 2, 2, 1, 3, 6, 3, 1, 2, 2, 3, 1, 1, 5, 30, 15, 5, 5, 1, 6, 2, 3, 1, 5, 1, 1, 7, 42, 21, 35, 35, 7, 7, 1, 2, 2, 15, 1, 5, 1, 7, 1, 1, 3, 30, 3, 5, 5, 7, 7, 1, 1, 1, 10, 2, 3, 1, 35, 1, 7, 1, 1, 1, 1, 11, 66, 165, 385, 55, 77, 77, 11, 11, 11, 11, 1

Offset: 0

Author

Peter Luschny, Apr 29 2009

Keywords

• nonn
• tabl

Comments

T(n,k) = Product_{ p - k | n} p, where p is prime.

T(n,0) is the squarefree kernel of n (A007947).

T(n,1) are the classical Clausen numbers (A141056). The classical Clausen numbers are by the von Staudt-Clausen theorem the denominators of the Bernoulli numbers.

Examples

			[k\n][0--1--2---3---4---5---6---7----8----9---10---11----12---13---14----15]
[0]...1..1..2...3...2...5...6...7....2....3...10...11.....6...13...14....15
[1]...1..2..6...2..30...2..42...2...30....2...66....2..2730....2....6.....2
[2]...1..3..3..15...3..21..15...3....3..165...21...39....15....3....3..1785
[3]...1..1..5...1..35...1...5...1..385....1...65....1....35....1...85.....1
[4]...1..5..5..35...5...5..35..55....5..455....5....5....35...85...55...665
[5]...1..1..7...1...7...1..77...1...91....1....7....1..1309....1..133.....1
T(3,4) = 35 = 5*7 because 5 and 7 are the only prime numbers p such that
(p - 4) divides 3.
		

References

• Clausen, Thomas, "Lehrsatz aus einer Abhandlung ueber die Bernoullischen Zahlen", Astr. Nachr. 17 (1840), 351-352.

Links

• Charles R Greathouse IV, Rows n = 0..100, flattened
• A. Hurwitz, Über die Entwicklungskoeffizienten der lemniskatischen Funktionen, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII.
• Peter Luschny, Generalized Bernoulli numbers.

Crossrefs

Cf. A007947, A141056, A027760, A027642.

Programs

Maple

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

Mathematica

t[0, ] = 1; t[n, k_] := Times @@ (Select[Divisors[n], PrimeQ[# + k] &] + k); Table[t[n-k, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 26 2013 *)

PARI

T(n,k)=if(n,my(s=1);fordiv(n,d,if(isprime(d+k),s*=d+k)); s, 1)
for(s=0,9,for(k=0,s,print1(T(s-k,k)", "))) \\ Charles R Greathouse IV, Jun 26 2013

Sage

def Clausen(n, k):
    if k == 0: return 1
    return mul(filter(lambda s: is_prime(s), map(lambda i: i+n, divisors(k))))
for n in (0..5): [Clausen(n, k) for k in (0..15)]   # Peter Luschny, Jun 05 2013

Extensions

Swapped n<>k fixed by Peter Luschny, May 04 2009