

A007674


Numbers n such that n and n+1 are squarefree.
(Formerly M1322)


22



1, 2, 5, 6, 10, 13, 14, 21, 22, 29, 30, 33, 34, 37, 38, 41, 42, 46, 57, 58, 61, 65, 66, 69, 70, 73, 77, 78, 82, 85, 86, 93, 94, 101, 102, 105, 106, 109, 110, 113, 114, 118, 122, 129, 130, 133, 137, 138, 141, 142, 145
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OFFSET

1,2


COMMENTS

n and n+1 squarefree implies n*(n+1) is squarefree oblong number, n*(n+1)/2 is squarefree triangular number.  Daniel Forgues, Aug 18 2012
Numbers n such that A002378(n) is squarefree.  Thomas Ordowski, Sep 01 2015


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

T. D. Noe, Table of n, a(n) for n = 1..1000
L. Carlitz, On a problem in additive arithmetic II, Quarterly Journal of Mathematics 3 (1932), pp. 273290.
Thomas Reuss, Pairs of kfree numbers, consecutive squarefull numbers, arXiv:1212.3150 [math.NT], 20122014.


FORMULA

A008966(a(n))*A008966(a(n)+1) = 1.  Reinhard Zumkeller, Dec 03 2009
a(n) ~ k*n, where k = 1/A065474. This result is originally due to Carlitz; for the (current) best error term, see Reuss.  Charles R Greathouse IV, Aug 10 2011, expanded Sep 18 2019


MATHEMATICA

ff = {}; gg = {}; Do[kk = FactorInteger[n]; tak = False; Do[If[kk[[m]][[2]] > 1, tak = True], {m, 1, Length[kk]}]; If[tak == False, jj = FactorInteger[n + 1]; tak1 = False; Do[If[jj[[m]][[2]] > 1, tak1 = True], {m, 1, Length[jj]}]; If[tak1 == False, AppendTo[ff, n]]], {n, 1, 500}]; ff (* Artur Jasinski, Jan 28 2010 *)
Select[Range[400], SquareFreeQ[#(#+1)]&] (* Vladimir Joseph Stephan Orlovsky, Mar 30 2011 *)


PROG

(PARI) list(lim)=my(v=vectorsmall(lim\1, i, 1), u=List()); for(n=2, sqrt(lim), forstep(i=n^2, lim, n^2, v[i]=v[i1]=0)); for(i=1, lim, if(v[i], listput(u, i))); v=0; Vec(u) \\ Charles R Greathouse IV, Aug 10 2011


CROSSREFS

Cf. A005117, A013929, A172186, A172187.
Sequence in context: A047440 A255055 A344314 * A086719 A115200 A075823
Adjacent sequences: A007671 A007672 A007673 * A007675 A007676 A007677


KEYWORD

nonn,easy


AUTHOR

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


EXTENSIONS

Initial 1 added at suggestion of Zak Seidov, Sep 19 2007


STATUS

approved



