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A112141
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Product of the first n semiprimes.
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22
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4, 24, 216, 2160, 30240, 453600, 9525600, 209563200, 5239080000, 136216080000, 4495130640000, 152834441760000, 5349205461600000, 203269807540800000, 7927522494091200000, 364666034728195200000, 17868635701681564800000, 911300420785759804800000
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OFFSET
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1,1
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COMMENTS
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Semiprime analog of primorial (A002110). Equivalent for product of what A062198 is for sum.
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LINKS
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FORMULA
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a(n) = Product_{i=1..n} A001358(i).
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EXAMPLE
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a(10) = 4*6*9*10*14*15*21*22*25*26 = 136216080000, the product of the first 10 semiprimes.
The sequence of terms together with their prime signatures begins:
4: (2)
24: (3,1)
216: (3,3)
2160: (4,3,1)
30240: (5,3,1,1)
453600: (5,4,2,1)
9525600: (5,5,2,2)
209563200: (6,5,2,2,1)
5239080000: (6,5,4,2,1)
136216080000: (7,5,4,2,1,1)
4495130640000: (7,6,4,2,2,1)
152834441760000: (8,6,4,2,2,1,1)
5349205461600000: (8,6,5,3,2,1,1)
203269807540800000: (9,6,5,3,2,1,1,1)
7927522494091200000: (9,7,5,3,2,2,1,1)
364666034728195200000: (10,7,5,3,2,2,1,1,1)
17868635701681564800000: (10,7,5,5,2,2,1,1,1)
(End)
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MAPLE
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end proc:
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MATHEMATICA
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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 *)
FoldList[Times, Select[Range[30], PrimeOmega[#]==2&]] (* Gus Wiseman, Dec 06 2020 *)
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PROG
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(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
(Python)
from sympy import factorint
def aupton(terms):
alst, k, p = [], 1, 1
while len(alst) < terms:
if sum(factorint(k).values()) == 2:
p *= k
alst.append(p)
k += 1
return alst
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CROSSREFS
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Partial sums of semiprimes are A062198.
First differences of semiprimes are A065516.
A000142 lists factorials, with partial products A000178 (superfactorials).
A001358 lists semiprimes, with partial products A112141 (this sequence).
A320655 counts factorizations into semiprimes.
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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