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A160014 Generalized Clausen numbers (table read by antidiagonals). 39
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 (list; table; graph; refs; listen; history; text; internal format)
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.
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
Sequence in context: A125231 A117919 A309106 * A068956 A124842 A134399
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Apr 29 2009
EXTENSIONS
Swapped n<>k fixed by Peter Luschny, May 04 2009
STATUS
approved

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Last modified March 19 02:55 EDT 2024. Contains 370952 sequences. (Running on oeis4.)