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A112141 Product of the first n semiprimes.

%I A112141 #30 Aug 31 2021 21:21:37
%S A112141 4,24,216,2160,30240,453600,9525600,209563200,5239080000,136216080000,
%T A112141 4495130640000,152834441760000,5349205461600000,203269807540800000,
%U A112141 7927522494091200000,364666034728195200000,17868635701681564800000,911300420785759804800000
%N A112141 Product of the first n semiprimes.
%C A112141 Semiprime analog of primorial (A002110). Equivalent for product of what A062198 is for sum.
%H A112141 T. D. Noe, <a href="/A112141/b112141.txt">Table of n, a(n) for n = 1..100</a>
%F A112141 a(n) = Product_{i=1..n} A001358(i).
%F A112141 A001222(a(n)) = 2*n.
%e A112141 a(10) = 4*6*9*10*14*15*21*22*25*26 = 136216080000, the product of the first 10 semiprimes.
%e A112141 From _Gus Wiseman_, Dec 06 2020: (Start)
%e A112141 The sequence of terms together with their prime signatures begins:
%e A112141                         4: (2)
%e A112141                        24: (3,1)
%e A112141                       216: (3,3)
%e A112141                      2160: (4,3,1)
%e A112141                     30240: (5,3,1,1)
%e A112141                    453600: (5,4,2,1)
%e A112141                   9525600: (5,5,2,2)
%e A112141                 209563200: (6,5,2,2,1)
%e A112141                5239080000: (6,5,4,2,1)
%e A112141              136216080000: (7,5,4,2,1,1)
%e A112141             4495130640000: (7,6,4,2,2,1)
%e A112141           152834441760000: (8,6,4,2,2,1,1)
%e A112141          5349205461600000: (8,6,5,3,2,1,1)
%e A112141        203269807540800000: (9,6,5,3,2,1,1,1)
%e A112141       7927522494091200000: (9,7,5,3,2,2,1,1)
%e A112141     364666034728195200000: (10,7,5,3,2,2,1,1,1)
%e A112141   17868635701681564800000: (10,7,5,5,2,2,1,1,1)
%e A112141 (End)
%p A112141 A112141 := proc(n)
%p A112141     mul(A001358(i),i=1..n) ;
%p A112141 end proc:
%p A112141 seq(A112141(n),n=1..10) ; # _R. J. Mathar_, Jun 30 2020
%t A112141 NextSemiPrime[n_, k_: 1] := Block[{c = 0, sgn = Sign[k]}, sp = n + sgn; While[c < Abs[k], While[ PrimeOmega[sp] != 2, If[sgn < 0, sp--, sp++]]; If[sgn < 0, sp--, sp++]; c++]; sp + If[sgn < 0, 1, -1]]; f[n_] := Times @@ NestList[ NextSemiPrime@# &, 2^2, n - 1]; Array[f, 18] (* _Robert G. Wilson v_, Jun 13 2013 *)
%t A112141 FoldList[Times,Select[Range[30],PrimeOmega[#]==2&]] (* _Gus Wiseman_, Dec 06 2020 *)
%o A112141 (PARI) a(n)=my(v=vector(n),i,k=3);while(i<n,if(bigomega(k++)==2,v[i++]=k)); prod(i=1,n,v[i]) \\ _Charles R Greathouse IV_, Apr 04 2013
%o A112141 (Python)
%o A112141 from sympy import factorint
%o A112141 def aupton(terms):
%o A112141     alst, k, p = [], 1, 1
%o A112141     while len(alst) < terms:
%o A112141         if sum(factorint(k).values()) == 2:
%o A112141             p *= k
%o A112141             alst.append(p)
%o A112141         k += 1
%o A112141     return alst
%o A112141 print(aupton(18)) # _Michael S. Branicky_, Aug 31 2021
%Y A112141 Partial sums of semiprimes are A062198.
%Y A112141 First differences of semiprimes are A065516.
%Y A112141 A000040 lists primes, with partial products A002110 (primorials).
%Y A112141 A000142 lists factorials, with partial products A000178 (superfactorials).
%Y A112141 A001358 lists semiprimes, with partial products A112141 (this sequence).
%Y A112141 A005117 lists squarefree numbers, with partial products A111059.
%Y A112141 A006881 lists squarefree semiprimes, with partial products A339191.
%Y A112141 A101048 counts partitions into semiprimes (restricted: A338902).
%Y A112141 A320655 counts factorizations into semiprimes.
%Y A112141 A338898/A338912/A338913 give the prime indices of semiprimes, with product/sum/difference A087794/A176504/A176506.
%Y A112141 A338899/A270650/A270652 give the prime indices of squarefree semiprimes, with product/sum/difference A339361/A339362/A338900.
%Y A112141 Cf. A001222, A084126, A084127, A115392, A168472, A320732, A320892.
%K A112141 easy,nonn
%O A112141 1,1
%A A112141 _Jonathan Vos Post_, Nov 28 2005