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A007675
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Numbers m such that m, m+1 and m+2 are squarefree.
(Formerly M3824)
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23
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1, 5, 13, 21, 29, 33, 37, 41, 57, 65, 69, 77, 85, 93, 101, 105, 109, 113, 129, 137, 141, 157, 165, 177, 181, 185, 193, 201, 209, 213, 217, 221, 229, 237, 253, 257, 265, 281, 285, 301, 309, 317, 321, 329, 345, 353, 357, 365, 381, 389, 393, 397, 401, 409, 417, 429, 433, 437, 445, 453
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OFFSET
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1,2
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COMMENTS
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Four categories: all terms are composites like {33, 34, 35}; first term only is prime like {37, 38, 39}; third term only is prime like {57, 58, 59}; first and third are primes like {29, 30, 31}. - Labos Elemer
Four consecutive integers cannot be squarefree as one of them is divisible by 2^2 = 4. - Amarnath Murthy, Feb 18 2002
Proof: m^3 + 3m^2 + 2m = m*(m+1)*(m+2) and the factors are pairwise relatively prime, because (m+1) is even. - Thomas Ordowski, Apr 20 2013
Conjecture: for every prime p, the numbers p# - 1, p#, p# + 1 are squarefree, where primorial p# = product of all primes <= p. - Thomas Ordowski, Apr 21 2013
Let f(m) = abs(mu(m*(m+1)*(m+2))), where mu(m) is the Moebius function, then the sum S(m) = f(1) + f(2) + ... + f(m) ~ k*m with the constant k = A206256 = 0.12548698.... - Thomas Ordowski, Apr 22 2013
All terms are congruent to 1 (mod 4). - Zak Seidov, Dec 22 2014
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REFERENCES
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P. R. Halmos, Problems for Mathematicians Young and Old. Math. Assoc. America, 1991, p. 28.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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Numbers m such that g(m)*g(m+1)*g(m+2) = 1, where g(w) = abs(mu(w)). - Labos Elemer
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EXAMPLE
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85 is a term as 85 = 17*5, 86 = 43*2, 87 = 29*3.
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MATHEMATICA
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Transpose[Select[Partition[Select[Range[400], SquareFreeQ], 3, 1], Differences[#] == {1, 1} &]][[1]] (* Harvey P. Dale, Apr 11 2012 *)
Select[Range[1, 499, 2], MoebiusMu[#^3 + 3#^2 + 2#] != 0 &] (* Alonso del Arte, Jan 16 2014 *)
SequencePosition[Table[If[SquareFreeQ[n], 1, 0], {n, 500}], {1, 1, 1}][[All, 1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Dec 14 2017 *)
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PROG
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(Haskell)
a007675 n = a007675_list !! (n-1)
a007675_list = f 1 a008966_list where
f n (u:xs'@(v:w:x:xs)) | u == 1 && w == 1 && v == 1 = n : f (n+4) xs
| otherwise = f (n+1) xs'
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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STATUS
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approved
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