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 A039833 Smallest of three consecutive squarefree numbers n, n+1, n+2 of the form p*q where p and q are primes. 8

%I

%S 33,85,93,141,201,213,217,301,393,445,633,697,921,1041,1137,1261,1345,

%T 1401,1641,1761,1837,1893,1941,1981,2101,2181,2217,2305,2361,2433,

%U 2461,2517,2641,2721,2733,3097,3385,3601,3693,3865,3901,3957,4285,4413,4533,4593,4881,5601

%N Smallest of three consecutive squarefree numbers n, n+1, n+2 of the form p*q where p and q are primes.

%C Equivalently: n, n+1 and n+2 all have 4 divisors.

%C There cannot be four consecutive squarefree numbers as one of them is divisible by 2^2 =4.

%C These 3 consecutive squarefree numbers of form pq have altogether 6 prime-factors always including 2 and 3. E.g. if n=99985, the six prime-factors are {2,3,5,19997,33329,49993}. The middle term is even and not divisible by 3.

%D D. Wells, Curious and interesting numbers, Penguin Books.

%H _Reinhard Zumkeller_, <a href="/A039833/b039833.txt">Table of n, a(n) for n = 1..1000</a>

%F A008966(a(n)) * A064911(a(n)) * A008966(a(n)+1) * A064911(a(n)+1) * A008966(a(n)+2) * A064911(a(n)+2) = 1. [_Reinhard Zumkeller_, Feb 26 2011]

%e 33, 34 and 35 all have 4 divisors. 85 is a term as 85 = 17*5, 86 = 43*2, 87 = 29*3.

%t lst = {}; Do[z = n^3 + 3*n^2 + 2*n; If[PrimeOmega[z/n] == PrimeOmega[z/(n + 2)] == 4 && PrimeNu[z] == 6, AppendTo[lst, n]], {n, 1, 5601, 2}]; lst (* Arkadiusz Wesolowski, Dec 11 2011 *)

%t okQ[n_]:=Module[{cl={n,n+1,n+2}},And@@SquareFreeQ/@cl && Union[ DivisorSigma[ 0,cl]]=={4}]; Select[Range[1,6001,2],okQ] (* From Harvey P. Dale, Dec 17 2011 *)

%o a039833 n = a039833_list !! (n-1)

%o a039833_list = f a006881_list where

%o f (u : vs@(v : w : xs))

%o | v == u+1 && w == v+1 = u : f vs

%o | otherwise = f vs

%o -- _Reinhard Zumkeller_, Aug 07 2011

%Y Cf. A038456, A039832, A008683, A007675, A063736, A063838, A070552, A045939, A056809.

%Y Subsequence of A006881.

%K nonn,nice

%O 1,1