A007674 Numbers m such that m and m+1 are squarefree.

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

Offset: 1

Author

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

Keywords

• nonn
• easy

Comments

m and m+1 squarefree implies that m*(m+1) is a squarefree oblong number and that m*(m+1)/2 is a squarefree triangular number. - Daniel Forgues, Aug 18 2012

Numbers m such that A002378(m) 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. 273-290.
• Thomas Reuss, Pairs of k-free numbers, consecutive square-full numbers, arXiv:1212.3150 [math.NT], 2012-2014.

Crossrefs

Cf. A005117, A013929, A172186, A172187.

Programs

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 *)

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

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

Extensions

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