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%I M1322 #49 Mar 22 2024 19:25:27
%S 1,2,5,6,10,13,14,21,22,29,30,33,34,37,38,41,42,46,57,58,61,65,66,69,
%T 70,73,77,78,82,85,86,93,94,101,102,105,106,109,110,113,114,118,122,
%U 129,130,133,137,138,141,142,145
%N Numbers m such that m and m+1 are squarefree.
%C 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
%C Numbers m such that A002378(m) is squarefree. - _Thomas Ordowski_, Sep 01 2015
%D P. R. Halmos, Problems for Mathematicians Young and Old. Math. Assoc. America, 1991, p. 28.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A007674/b007674.txt">Table of n, a(n) for n = 1..1000</a>
%H L. Carlitz, <a href="https://doi.org/10.1093/qmath/os-3.1.273">On a problem in additive arithmetic II</a>, Quarterly Journal of Mathematics 3 (1932), pp. 273-290.
%H Thomas Reuss, <a href="https://arxiv.org/abs/1212.3150">Pairs of k-free numbers, consecutive square-full numbers</a>, arXiv:1212.3150 [math.NT], 2012-2014.
%F A008966(a(n))*A008966(a(n)+1) = 1. - _Reinhard Zumkeller_, Dec 03 2009
%F 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
%t 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 *)
%t Select[Range[400],SquareFreeQ[#(#+1)]&] (* _Vladimir Joseph Stephan Orlovsky_, Mar 30 2011 *)
%o (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
%Y Cf. A005117, A013929, A172186, A172187.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_, _Robert G. Wilson v_
%E Initial 1 added at the suggestion of _Zak Seidov_, Sep 19 2007
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