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

0,4

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.

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.

Example

[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.

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 *)

Prog

(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
(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

Crossrefs

Cf. A007947, A141056, A027760, A027642.

Sequence in context: A125231 A117919 A309106 * A068956 A124842 A134399

Adjacent sequences: A160011 A160012 A160013 * A160015 A160016 A160017

Keyword

nonn,tabl

Author

Peter Luschny, Apr 29 2009

Extensions

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

Status

approved