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%I #40 Jan 19 2019 04:14:59
%S 33,1309,203433,16467033,1990586013,41704979953,102099792179229
%N For any integer m > 1, the m-th term of the sequence is the minimal squarefree integer x with exactly m prime divisors such that x+1 and x+2 are also squarefree integers with exactly m prime divisors.
%C The five terms for m = 2,3,4,5,6 were computed with the aid of PARI/GP. But it seems to be rather difficult to compute higher terms, if they exist at all.
%C The distribution of squarefree integers with exactly m prime factors is given in the book by Montgomery and Vaughan, Multiplicative Number Theory, but I do not have access to it and do not know whether it also addresses the problem of three consecutive numbers of this kind.
%D Hugh L. Montgomery and Robert C. Vaughan: "Multiplicative Number Theory: 1. Classical Theory", Cambridge studies in advanced mathematics, vol. 97, Cambridge University Press (2007)
%H Math Overflow, <a href="http://mathoverflow.net/questions/160963/asymptotics-of-special-square-free-numbers">Asymptotics of special square-free numbers</a>, Mar 20 2014
%H Daniel Constantin Mayer, <a href="/A242492/a242492.gp.txt">PARI/GP script "AdjacentSquareFree.gp"</a>
%H Daniel Constantin Mayer, <a href="/A242492/a242492_1.gp.txt">PARI/GP script "SquareFreeTriplets.gp"</a>
%H Prime puzzles & problems connection, <a href="http://www.primepuzzles.net/puzzles/puzz_371.htm">Square-free triples</a>
%F a(n) = A093550(n)-1. - _M. F. Hasler_, May 20 2014
%e 33 = 3*11, 34 = 2*17, 35 = 5*7;
%e 1309 = 7*11*17, 1310 = 2*5*131, 1311 = 3*19*23;
%e 203433 = 3*19*43*83, 203434 = 2*7*11*1321, 203435 = 5*23*29*61;
%e 16467033 = 3*11*17*149*197, 16467034 = 2*19*23*83*227, 16467035 = 5*13*37*41*167; (CPU time 48 seconds)
%e 1990586013 = 3*13*29*67*109*241, 1990586014 = 2*23*37*43*59*461, 1990586015 = 5*11*17*19*89*1259. (CPU time 2 hours and 34 minutes)
%o (PARI)
%o {default(primelimit,2M); lb=2; ub=2*10^9; m=1; i=0; j=0; loc=0; while(m<6, m=m+1; for(n=lb,ub, if(issquarefree(n)&&(m==omega(n)), loc=loc+1; if(1==loc, i=n; ); if(2==loc, if(i+1==n, j=n; ); if(i+1<n, loc=1; i=n; ); ); if(3==loc, if(j+1==n, print(i); lb=i+1; loc=0; break(1); ); if(j+1<n, loc=1; i=n; ); ); ); ); ); }
%o \\ For more detailed information, execute the PARI scripts
%o \\ "SquareFreeTriplets.gp", resp. "AdjacentSquareFree.gp",
%o \\ in the linked supporting files.
%Y Cf. A007675 (any m), A039833 (m=2), A066509 (m=3), A176167 (m=4), A192203 (m=5), A068088 (sextets with gap).
%Y Cf. A242605-A242608 for start of triples of consecutive squarefree numbers with m=2,...,5 prime factors, A242621 for the analog of the present sequence in that spirit.
%K hard,nonn,more
%O 2,1
%A _Daniel Constantin Mayer_, May 16 2014
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