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A039833
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Smallest of three consecutive squarefree numbers n, n+1, n+2 of the form p*q where p and q are primes.
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9
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33, 85, 93, 141, 201, 213, 217, 301, 393, 445, 633, 697, 921, 1041, 1137, 1261, 1345, 1401, 1641, 1761, 1837, 1893, 1941, 1981, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2721, 2733, 3097, 3385, 3601, 3693, 3865, 3901, 3957, 4285, 4413, 4533, 4593, 4881, 5601
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Equivalently: n, n+1 and n+2 all have 4 divisors.
There cannot be four consecutive squarefree numbers as one of them is divisible by 2^2 =4.
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.
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REFERENCES
| D. Wells, Curious and interesting numbers, Penguin Books.
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LINKS
| Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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FORMULA
| 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]
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EXAMPLE
| 33, 34 and 35 all have 4 divisors. 85 is a term as 85 = 17*5, 86 = 43*2, 87 = 29*3.
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MATHEMATICA
| 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 *)
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 *)
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PROG
| (Haskell)
a039833 n = a039833_list !! (n-1)
a039833_list = f a006881_list where
f (u : vs@(v : w : xs))
| v == u+1 && w == v+1 = u : f vs
| otherwise = f vs
-- Reinhard Zumkeller, Aug 07 2011
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CROSSREFS
| Cf. A038456, A039832, A008683, A007675, A063736, A063838, A070552, A045939, A056809.
Subsequence of A006881.
Sequence in context: A052214 A063838 A075039 * A080700 A080200 A067705
Adjacent sequences: A039830 A039831 A039832 * A039834 A039835 A039836
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KEYWORD
| nonn,nice
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AUTHOR
| Olivier Gerard (olivier.gerard(AT)gmail.com)
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EXTENSIONS
| Additional comments from Amarnath Murthy, Vladeta Jovovic, Labos E. and Benoit Cloitre, May 08 2002
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