OFFSET
1,2
COMMENTS
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
Numbers m such that m^3 + 3m^2 + 2m is squarefree. See proof below. - Charles R Greathouse IV, Mar 05 2013
There are kx + O(x/log x) terms of this sequence below x, where k = A206256. - Charles R Greathouse IV, Mar 05 2013
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
REFERENCES
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).
LINKS
Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
FORMULA
Numbers m such that g(m)*g(m+1)*g(m+2) = 1, where g(w) = abs(mu(w)). - Labos Elemer
a(n) ~ c*n with c = 7.96895... = 1/A206256. - Charles R Greathouse IV, Mar 05 2013
EXAMPLE
85 is a term as 85 = 17*5, 86 = 43*2, 87 = 29*3.
MAPLE
select(t -> andmap(NumberTheory:-IsSquareFree, [t, t+1, t+2]), [seq(i, i=1..1000, 4)]); # Robert Israel, Jul 16 2024
MATHEMATICA
Select[Range[1000], SquareFreeQ[#(# + 1)(# + 2)] &] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011 *)
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 *)
PROG
(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'
-- Reinhard Zumkeller, Nov 05 2011
(PARI) is(n)=issquarefree(n)&&issquarefree(n+1)&&issquarefree(n+2) \\ Charles R Greathouse IV, Mar 05 2013
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
STATUS
approved