A242793 - OEIS

%I #20 Jan 19 2019 04:14:59

%S 213,73293,9743613,6639266409

%N The minimal integer x such that each of the six integers x, x+1, x+2, x+4, x+5, x+6 is squarefree with exactly n prime divisors.

%C This is the next step in my project to study the distribution of increasingly extensive clusters of squarefree integers with fixed number of prime divisors: triples x,x+1,x+2 were investigated in A242492 and here we study sextets x,x+1,x+2,x+4,x+5,x+6 with a central gap x+3, since x+3 must be divisible by the square 4.

%C The term 6639266409 required 30 hours of CPU time on an iMac with Intel i7 Quadcore CPU running OS X Lion.

%e 213=3*71, 214=2*107, 215=5*43, 217=7*31, 218=2*109, 219=3*73;

%e 73293=3*11*2221, 73294=2*13*2819, 73295=5*107*137,

%e 73297=7*37*283, 73298=2*67*547, 73299=3*53*461;

%e 9743613=3*11*503*587, 9743614=2*59*71*1163, 9743615= 5*7*167*1667, 9743617=13*37*47*431, 9743618=2*17*19*15083, 9743619=3*83*109*359;

%e 6639266409=3*29*109*421*1663, 6639266410=2*5*7*113*839351,

%e 6639266411=17*23*89*101*1889, 6639266413=13*61*79*131*809,

%e 6639266414=2*11*349*857*1009, 6639266415=3*5*73*149*40693.

%o (PARI)

%o { default(primelimit,1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=1; while(o<5, o=o+1; for(n=lb,ub, if(issquarefree(n)&&(o==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, k=n; ); if(j+1<n, loc=1; i=n; ); ); if(4==loc, if(k+2==n, l=n; ); if(k+2<n, loc=1; i=n; ); ); if(5==loc, if(l+1==n, m=n; ); if(l+1<n, loc=1; i=n; ); ); if(6==loc, if(m+1==n, print1(i,","); lb=i+1; loc=0; break(1); ); if(m+1<n, loc=1; i=n; ); ); ); ); ); }

%Y Cf. A242492 (triples) and A242804, A242805, A242806, A242829, and also A068088.

%K hard,more,nonn

%O 2,1

%A _Daniel Constantin Mayer_, May 22 2014