

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


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



FORMULA

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 *)


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



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS

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


STATUS

approved



