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

1, 15, 35, 143, 225, 323, 525, 899, 1225, 1763, 2145, 3375, 3599, 4845, 5005, 5183, 7875, 10403, 11305, 11663, 13485, 18375, 19043, 20449, 22499, 26445, 31465, 32175, 32399, 36863, 39203, 42875, 46189, 50625, 51983, 53985, 57599, 61705

Offset: 1

Author

Reinhard Zumkeller, Aug 23 2002

Keywords

• nonn
• nice

Examples

			a(99) = 1040399 = 1019*1021.
a(101) = 1090125 = (3*5)*(3*5)*(3*5)*(17*19).
a(103) = 1101275 = (5*7)*(5*7)*(29*31).
a(105) = 1126125 = (3*5)*(3*5)*(5*7)*(11*13).
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A037074, A001359, A006512, A062505, A072965.

Cf. A071700 (subsequence).

Programs

Haskell

import Data.Set (Set, singleton, delete, findMin, deleteFindMin, insert)
a074480 n = a074480_list !! (n-1)
a074480_list = multClosure a037074_list where
  multClosure []     = [1]
  multClosure (b:bs) = 1:h [b] (singleton b) bs where
   h cs s []    = m:h (m:cs) (foldl (flip insert) s' $ map (*m) cs) []
    where (m, s') = deleteFindMin s
   h cs s xs'@(x:xs)
    | m < x     = m:h (m:cs) (foldl (flip insert) s' $ map (*m) cs) xs'
    | otherwise = x:h (x:cs) (foldl (flip insert) s  $ map (*x) (x:cs)) xs
    where (m, s') = deleteFindMin s
-- Reinhard Zumkeller, Aug 14 2011

Mathematica

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 *)

Formula

A072965(a(n)) = 1.

Sum_{n>=1} 1/a(n) = 1/Product_{k>=1} (1 - 1/A037074(k)) = 1.117122860973... . - Amiram Eldar, Apr 13 2025