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A007674 Numbers m such that m and m+1 are squarefree.
(Formerly M1322)
36
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
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
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.
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[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
Sequence in context: A358817 A369166 A369211 * A086719 A115200 A075823
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Initial 1 added at the suggestion of Zak Seidov, Sep 19 2007
STATUS
approved

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Last modified May 23 02:40 EDT 2024. Contains 372758 sequences. (Running on oeis4.)