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A114411 Triple primorial n### = n#3.

%I A114411 #13 Feb 16 2025 08:32:59
%S A114411 1,2,3,5,14,33,65,238,627,1495,6902,19437,55315,282982,835791,2599805,
%T A114411 14998046,49311669,158588105,1004869082,3501128499,11576931665,
%U A114411 79384657478,290593665417,1030346918185,7700311775366,29349960207117
%N A114411 Triple primorial n### = n#3.
%C A114411 This is to triple factorial A007661 = n!!!, as double primorial A079078 = n## is to double factorial A006882 = n!! and as primorial A002110 = n# is to factorial A000142 = n!. There is an obvious generalization to multiprimorial. (n###)*((n-1)###)*((n-2)###) = n#. n### is a k-almost prime for k = ceiling(n/3).
%H A114411 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Primorial.html">Primorial</a>.
%H A114411 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Multifactorial.html">Multifactorial</a>.
%F A114411 a(n) = n### = prime(n)*((n-3)###) = Prod[i == n mod 3, to n] prime(i). Notationally, prime(0) = 1; (-n)### = 0### = 1.
%e A114411 n### is also written n#3.
%e A114411 0### = p(0) = 1.
%e A114411 1### = p(1) = 2.
%e A114411 2### = p(2) = 3.
%e A114411 3### = p(3)p(0) = 5*1 = 5.
%e A114411 4### = p(4)p(1) = 7*2 = 14.
%e A114411 5### = p(5)p(2) = 11*3 = 33.
%e A114411 6### = p(6)p(3)p(0) = 13*5*1 = 65.
%e A114411 7### = p(7)p(4)p(1) = 17*7*2 = 238.
%e A114411 8### = p(8)p(5)p(2) = 19*11*3 = 627.
%e A114411 9### = p(9)p(6)p(3)p(0) = 23*13*5*1 = 1495.
%e A114411 10### = p(10)p(7)p(4)p(1) = 29*17*7*2 = 6902.
%e A114411 11### = p(11)p(8)p(5)p(2) = 31*19*11*3 = 19437.
%e A114411 12### = 37*23*13*5*1 = 55315.
%e A114411 13### = 41*29*17*7*2 = 282982.
%e A114411 14### = 43*31*19*11*3 = 835791.
%e A114411 15### = 47*37*23*13*5*1 = 2599805.
%e A114411 27### = 106125732573055 = 5 * 13 * 23 * 37 * 47 * 61 * 73 * 89 * 103.
%t A114411 a[0] = 1; a[1] = 2; a[2] = 3;
%t A114411 a[n_] := a[n - 3] * Prime[n]
%t A114411 Array[a, 27, 0] (* _Jon Maiga_, Aug 04 2019 *)
%Y A114411 Cf. A000142, A002110, A006882, A007661, A079078.
%K A114411 nonn
%O A114411 0,2
%A A114411 _Jonathan Vos Post_, Feb 12 2006