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A160014 Generalized Clausen numbers (table read by antidiagonals).

%I A160014 #25 Aug 11 2025 08:02:12
%S A160014 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,
%T A160014 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,
%U A160014 1,1,11,66,165,385,55,77,77,11,11,11,11,1
%N A160014 Generalized Clausen numbers (table read by antidiagonals).
%C A160014 T(n,k) = Product_{ p - k | n} p, where p is prime.
%C A160014 T(n,0) is the squarefree kernel of n (A007947).
%C A160014 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.
%D A160014 Clausen, Thomas, "Lehrsatz aus einer Abhandlung ueber die Bernoullischen Zahlen", Astr. Nachr. 17 (1840), 351-352.
%H A160014 Charles R Greathouse IV, <a href="/A160014/b160014.txt">Rows n = 0..100, flattened</a>
%H A160014 A. Hurwitz, <a href="https://gdz.sub.uni-goettingen.de/id/PPN235181684_0051">Über die Entwicklungskoeffizienten der lemniskatischen Funktionen</a>, Math. Ann., 51 (1899), 196-226; Mathematische Werke. Vols. 1 and 2, Birkhäuser, Basel, 1962-1963, see Vol. 2, No. LXVII.
%H A160014 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedBernoulliNumbers.html">Generalized Bernoulli numbers</a>.
%e A160014 [k\n][0--1--2---3---4---5---6---7----8----9---10---11----12---13---14----15]
%e A160014 [0]...1..1..2...3...2...5...6...7....2....3...10...11.....6...13...14....15
%e A160014 [1]...1..2..6...2..30...2..42...2...30....2...66....2..2730....2....6.....2
%e A160014 [2]...1..3..3..15...3..21..15...3....3..165...21...39....15....3....3..1785
%e A160014 [3]...1..1..5...1..35...1...5...1..385....1...65....1....35....1...85.....1
%e A160014 [4]...1..5..5..35...5...5..35..55....5..455....5....5....35...85...55...665
%e A160014 [5]...1..1..7...1...7...1..77...1...91....1....7....1..1309....1..133.....1
%e A160014 T(3,4) = 35 = 5*7 because 5 and 7 are the only prime numbers p such that
%e A160014 (p - 4) divides 3.
%p A160014 Clausen := proc(n,k) local S,i;
%p A160014 S := numtheory[divisors](n);
%p A160014 S := map(i->i+k,S);
%p A160014 S := select(isprime,S);
%p A160014 mul(i,i=S) end:
%t A160014 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 *)
%o A160014 (Sage)
%o A160014 def Clausen(n, k):
%o A160014     if k == 0: return 1
%o A160014     return mul(filter(lambda s: is_prime(s), map(lambda i: i+n, divisors(k))))
%o A160014 for n in (0..5): [Clausen(n, k) for k in (0..15)]   # _Peter Luschny_, Jun 05 2013
%o A160014 (PARI) T(n,k)=if(n,my(s=1);fordiv(n,d,if(isprime(d+k),s*=d+k)); s, 1)
%o A160014 for(s=0,9,for(k=0,s,print1(T(s-k,k)", "))) \\ _Charles R Greathouse IV_, Jun 26 2013
%Y A160014 Cf. A007947, A141056, A027760, A027642.
%K A160014 nonn,tabl
%O A160014 0,4
%A A160014 _Peter Luschny_, Apr 29 2009
%E A160014 Swapped n<>k fixed by _Peter Luschny_, May 04 2009