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A242804 Integers k such that each of k, k+1, k+2, k+4, k+5, k+6 is the product of two distinct primes.

%I A242804 #72 Jul 14 2021 01:56:17
%S A242804 213,143097,194757,206133,273417,684897,807657,1373937,1391757,
%T A242804 1516533,1591593,1610997,1774797,1882977,1891761,2046453,2051493,
%U A242804 2163417,2163957,2338053,2359977,2522517,2913837,3108201,4221753
%N A242804 Integers k such that each of k, k+1, k+2, k+4, k+5, k+6 is the product of two distinct primes.
%C A242804 A remarkable gap occurs between the initial two members, and the sequence seems to be rather sparse compared to the related A242805.
%C A242804 Here, the first member k of the sextet is the reference, whereas in A068088 the center k+3 is selected as reference. Observe that k+3 must be divisible by the square 4.
%C A242804 All terms are congruent to 9 (mod 12). - _Zak Seidov_, Apr 14 2015
%C A242804 From _Robert Israel_, Apr 15 2015: (Start)
%C A242804 All terms are congruent to 33 (mod 36).
%C A242804 Numbers k in A039833 such that k+4 is in A039833. (End)
%C A242804 From _Robert G. Wilson v_, Apr 15 2015: (Start)
%C A242804 k is congruent to 33 (mod 36) so one of its factors is 3 and the other is == 11 (mod 12);
%C A242804 k+1 is congruent to 34 (mod 36) so one of its factors is 2 and the other is == 17 (mod 18);
%C A242804 k+2 is congruent to 35 (mod 36) so its factors are == +-1 (mod 6);
%C A242804 k+4 is congruent to 1 (mod 36) so its factors are == +-1 (mod 6);
%C A242804 k+5 is congruent to 2 (mod 36) so one of its factors is 2 and the other is == 1 (mod 18);
%C A242804 k+6 is congruent to 3 (mod 36) so one of its factors is 3 and the other is == 1 (mod 12). (End).
%C A242804 Number of terms < 10^m: 0, 0, 1, 1, 1, 7, 39, 169, 882, 4852, 27479, ...,. - _Robert G. Wilson v_, Apr 15 2015
%C A242804 Or, numbers k such that k, k+1 and k+2 are terms in A175648. - _Zak Seidov_, Dec 08 2015
%H A242804 Zak Seidov and Robert G. Wilson v, <a href="/A242804/b242804.txt">Table of n, a(n) for n = 1..10000</a>
%F A242804 a(n) = A202319(n) - 1. - _Jon Maiga_, Jul 10 2021
%e A242804 213=3*71, 214=2*107, 215=5*43, 217=7*31, 218=2*109, 219=3*73.
%p A242804 f:= t -> numtheory:-issqrfree(t) and (numtheory:-bigomega(t) = 2):
%p A242804 select(t -> andmap(f, [t,t+1,t+2,t+4,t+5,t+6]), [seq(36*k+33,k=0..10^6)]); # _Robert Israel_, Apr 15 2015
%t A242804 fQ[n_] := PrimeQ[n/3] && PrimeQ[(n + 1)/2] && PrimeQ[(n + 5)/2] && PrimeQ[(n + 6)/3] && PrimeNu[{n + 2, n + 4}] == {2, 2} == PrimeOmega[{n + 2, n + 4}]; k = 33; lst = {}; While[k < 10^8, If[fQ@ k, AppendTo[lst, k]]; k += 36]; lst (* _Robert G. Wilson v_, Apr 14 2015 and revised Apr 15 2015 after _Zak Seidov_ and _Robert Israel_ *)
%o A242804 (PARI) default(primelimit, 1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=2; 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,","); loc=0; ); if(m+1<n, loc=1; i=n))))
%o A242804 (PARI) forstep(x=213,4221753,12, if( isprime(x/3) && isprime((x+1)/2) && 2==omega(x+2) && 2==bigomega(x+2) && 2==omega(x+4) && 2==bigomega(x+4) && isprime((x+5)/2) && isprime((x+6)/3), print1(x", "))) \\ _Zak Seidov_, Apr 14 2015
%Y A242804 Cf. A242793 (minima for two, three and more prime divisors) and A068088 (arbitrary squarefree integers).
%Y A242804 Cf. A242805, A039833, A175648, A202319.
%K A242804 nonn
%O A242804 1,1
%A A242804 _Daniel Constantin Mayer_, May 23 2014