

A242492


For any integer m > 1, the mth 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.


2




OFFSET

2,1


COMMENTS

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.
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.


REFERENCES

Hugh L. Montgomery and Robert C. Vaughan: "Multiplicative Number Theory: 1. Classical Theory", Cambridge studies in advanced mathematics, vol. 97, Cambridge University Press (2007)


LINKS

Table of n, a(n) for n=2..8.
Math Overflow, Asymptotics of special squarefree numbers, Mar 20 2014
Daniel Constantin Mayer, PARI/GP script "AdjacentSquareFree.gp"
Daniel Constantin Mayer, PARI/GP script "SquareFreeTriplets.gp"
Prime puzzles & problems connection, Squarefree triples


FORMULA

a(n) = A093550(n)1.  M. F. Hasler, May 20 2014


EXAMPLE

33 = 3*11, 34 = 2*17, 35 = 5*7;
1309 = 7*11*17, 1310 = 2*5*131, 1311 = 3*19*23;
203433 = 3*19*43*83, 203434 = 2*7*11*1321, 203435 = 5*23*29*61;
16467033 = 3*11*17*149*197, 16467034 = 2*19*23*83*227, 16467035 = 5*13*37*41*167; (CPU time 48 seconds)
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)


PROG

(PARI)
{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; ); ); ); ); ); }
\\ For more detailed information, execute the PARI scripts
\\ "SquareFreeTriplets.gp", resp. "AdjacentSquareFree.gp",
\\ in the linked supporting files.


CROSSREFS

Cf. A007675 (any m), A039833 (m=2), A066509 (m=3), A176167 (m=4), A192203 (m=5), A068088 (sextets with gap).
Cf. A242605A242608 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.
Sequence in context: A077420 A158688 A294436 * A065424 A071268 A012805
Adjacent sequences: A242489 A242490 A242491 * A242493 A242494 A242495


KEYWORD

hard,nonn,more


AUTHOR

Daniel Constantin Mayer, May 16 2014


STATUS

approved



