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
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
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
Comments