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A368093 Cumulative products of the generalized Clausen numbers. Array read by ascending antidiagonals.

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%I A368093 #9 Dec 14 2023 16:28:33
%S A368093 1,1,1,1,2,2,1,3,12,6,1,1,9,24,12,1,5,5,135,720,60,1,1,25,5,405,1440,
%T A368093 360,1,7,7,875,175,8505,60480,2520,1,1,49,7,4375,175,127575,120960,
%U A368093 5040,1,1,1,343,49,21875,875,382725,3628800,15120
%N A368093 Cumulative products of the generalized Clausen numbers. Array read by ascending antidiagonals.
%C A368093 A160014 are the generalized Clausen numbers, for m = 0 the formula computes the cumulative radical A048803, and for m = 1 the Hirzebruch numbers A091137.
%F A368093 A(m, n) = A160014(m, n) * A(m, n - 1) for n > 0 and A(m, 0) = 1.
%e A368093 Array A(m, n) starts:
%e A368093   [0] 1, 1,  2,   6,   12,     60,     360,      2520, ...  A048803
%e A368093   [1] 1, 2, 12,  24,  720,   1440,   60480,    120960, ...  A091137
%e A368093   [2] 1, 3,  9, 135,  405,   8505,  127575,    382725, ...  A368092
%e A368093   [3] 1, 1,  5,   5,  175,    175,     875,       875, ...
%e A368093   [4] 1, 5, 25, 875, 4375,  21875,  765625,  42109375, ...
%e A368093   [5] 1, 1,  7,   7,   49,     49,    3773,      3773, ...
%e A368093   [6] 1, 7, 49, 343, 2401, 184877, 1294139, 117766649, ...
%e A368093   [7] 1, 1,  1,   1,   11,     11,     143,       143, ...
%e A368093   [8] 1, 1,  1,  11,   11,    143,    1573,      1573, ...
%e A368093   [9] 1, 1, 11,  11, 1573,   1573,   17303,     17303, ...
%o A368093 (SageMath)
%o A368093 from functools import cache
%o A368093 def Clausen(n, k):
%o A368093     return mul(s for s in map(lambda i: i+n, divisors(k)) if is_prime(s))
%o A368093 @cache
%o A368093 def CumProdClausen(m, n):
%o A368093     return Clausen(m, n) * CumProdClausen(m, n - 1) if n > 0 else 1
%o A368093 for m in range(10): print([m], [CumProdClausen(m, n) for n in range(8)])
%Y A368093 Cf. A160014, A048803 (m=0), A091137 (m=1), A368092 (m=2).
%Y A368093 Cf. A171080, A238963, A368116, A368048.
%K A368093 nonn,tabl
%O A368093 0,5
%A A368093 _Peter Luschny_, Dec 12 2023