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A074480 Multiplicative closure of twin prime pair products (A037074).

%I A074480 #21 Apr 13 2025 03:04:00
%S A074480 1,15,35,143,225,323,525,899,1225,1763,2145,3375,3599,4845,5005,5183,
%T A074480 7875,10403,11305,11663,13485,18375,19043,20449,22499,26445,31465,
%U A074480 32175,32399,36863,39203,42875,46189,50625,51983,53985,57599,61705
%N A074480 Multiplicative closure of twin prime pair products (A037074).
%H A074480 Reinhard Zumkeller, <a href="/A074480/b074480.txt">Table of n, a(n) for n = 1..1000</a>
%F A074480 A072965(a(n)) = 1.
%F A074480 Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/A037074(k)) = 1.117122860973... . - _Amiram Eldar_, Apr 13 2025
%e A074480 a(99) = 1040399 = 1019*1021.
%e A074480 a(101) = 1090125 = (3*5)*(3*5)*(3*5)*(17*19).
%e A074480 a(103) = 1101275 = (5*7)*(5*7)*(29*31).
%e A074480 a(105) = 1126125 = (3*5)*(3*5)*(5*7)*(11*13).
%t A074480 max = 70000; t1 = Select[Prime /@ Range[PrimePi[Sqrt[max]]], PrimeQ[# + 2] &]; pairs = Join[{1}, t1*(t1 + 2)]; f[pairs_] := Outer[Times, pairs, pairs] // Flatten // Union // Select[#, # <= max &] &; FixedPoint[f, pairs] (* _Jean-François Alcover_, Dec 11 2012 *)
%o A074480 (Haskell)
%o A074480 import Data.Set (Set, singleton, delete, findMin, deleteFindMin, insert)
%o A074480 a074480 n = a074480_list !! (n-1)
%o A074480 a074480_list = multClosure a037074_list where
%o A074480   multClosure []     = [1]
%o A074480   multClosure (b:bs) = 1:h [b] (singleton b) bs where
%o A074480    h cs s []    = m:h (m:cs) (foldl (flip insert) s' $ map (*m) cs) []
%o A074480     where (m, s') = deleteFindMin s
%o A074480    h cs s xs'@(x:xs)
%o A074480     | m < x     = m:h (m:cs) (foldl (flip insert) s' $ map (*m) cs) xs'
%o A074480     | otherwise = x:h (x:cs) (foldl (flip insert) s  $ map (*x) (x:cs)) xs
%o A074480     where (m, s') = deleteFindMin s
%o A074480 -- _Reinhard Zumkeller_, Aug 14 2011
%Y A074480 Cf. A037074, A001359, A006512, A062505, A072965.
%Y A074480 Cf. A071700 (subsequence).
%K A074480 nonn,nice
%O A074480 1,2
%A A074480 _Reinhard Zumkeller_, Aug 23 2002