A007675 Numbers m such that m, m+1 and m+2 are squarefree.

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

Offset: 1

Author

N. J. A. Sloane, Robert G. Wilson v

Keywords

• nonn
• easy
• nice

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

Examples

			85 is a term as 85 = 17*5, 86 = 43*2, 87 = 29*3.
		

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)

Crossrefs

Subsequence of A007674, A016813, and A056911.

Cf. A005117, A013929, A008966, A007531.

Programs

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

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 *)

PARI

is(n)=issquarefree(n)&&issquarefree(n+1)&&issquarefree(n+2) \\ Charles R Greathouse IV, Mar 05 2013

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