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 A160014 Generalized Clausen numbers (table read by antidiagonals). 18
 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. 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: A080955 A125231 A117919 * 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

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Last modified March 22 19:08 EDT 2018. Contains 301083 sequences. (Running on oeis4.)