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A007674 Numbers n such that n and n+1 are squarefree.
(Formerly M1322)
14
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 (list; graph; refs; listen; history; text; internal format)
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

FORMULA

A008966(a(n))*A008966(a(n)+1) = 1. - Reinhard Zumkeller, Dec 03 2009

a(n) ~ k*n, where k = 1/A065474. - Charles R Greathouse IV, Aug 10 2011

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[i-1]=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: A089269 A047440 A255055 * 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

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Last modified April 20 16:17 EDT 2019. Contains 322310 sequences. (Running on oeis4.)