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A242805 Integers n such that each of n, n + 1, n + 2, n + 4, n + 5, n + 6 is the squarefree product of three primes.

%I A242805 #41 Jan 19 2019 03:38:16
%S A242805 73293,120237,122613,130429,143493,147953,171893,180965,199833,213153,
%T A242805 219201,268017,287493,298433,299553,300093,313701,329793,332889,
%U A242805 341781,363597,369393,376201,392509,404453,432393,460801,475809,493597,503457,506517,508677
%N A242805 Integers n such that each of n, n + 1, n + 2, n + 4, n + 5, n + 6 is the squarefree product of three primes.
%C A242805 It is remarkable that this sequence starts with considerably bigger density than the analog A242804 for squarefree integers with two prime divisors. The exceptional density causes the problem that overlapping sextets appear very soon and rather frequently, whereas in A242804 the phenomenon of overlapping sextets does not occur up to the bound 9*10^9.
%C A242805 In fact, there exist 114 nonets n, n + 1, n + 2, n + 4, n + 5, n + 6, n + 8, n + 9, n + 10 of squarefree integers with exactly three prime divisors, up to 10^8. The PARI script in PROG does not start a new sextet before the previous sextet was completed. The impact on bigger clusters, such as nonets and dodekuplets, is illustrated in the CAVEAT of section EXAMPLE.
%H A242805 David A. Corneth, <a href="/A242805/b242805.txt">Table of n, a(n) for n = 1..10000</a> (first 176 terms from Zak Seidov)
%e A242805 73293 = 3*11*2221, 73294 = 2*13*2819, 73295 = 5*107*137,
%e A242805 73297 = 7*37*283,  73298 = 2*67*547,  73299 = 3*53*461.
%e A242805 CAVEAT:
%e A242805 (1) For the dodekuplet, which starts together with the first nonet,
%e A242805 969833 = 17*89*641,  969834 = 2*3*161639, 969835 = 5*31*6257,
%e A242805 969837 = 3*11*29389, 969838 = 2*173*2803, 969839 = 13*61*1223,
%e A242805 969841 = 23*149*283, 969842 = 2*59*8219,  969843 = 3*7*46183,
%e A242805 969845 = 5*47*4127,  969846 = 2*3*161641, 969847 = 29*53*631.
%e A242805 Not all PARI scripts list 969833 and 969841, but not 969837.
%e A242805 (2) For the second nonet,
%e A242805 1450257 = 3*229*2111, 1450258 = 2*179*4051, 1450259 = 83*101*173,
%e A242805 1450261 = 29*43*1163, 1450262 = 2*11*65921, 1450263 = 3*191*2531,
%e A242805 1450265 = 5*23*12611, 1450266 = 2*3*241711, 1450267 = 7*13*15937,
%e A242805 the PARI script lists 1450257 only, but not 1450261.
%t A242805 s = {}; Do[If[AllTrue[{k, k + 1, k + 2, k + 4, k + 5, k + 6}, SquareFreeQ] && {3, 3, 3, 3, 3, 3} == PrimeOmega[{k, k + 1, k + 2, k + 4, k + 5, k + 6}], AppendTo[s, k]], {k, 73293, 2000000, 4}]; s (* _Zak Seidov_, Nov 12 2018 *)
%o A242805 (PARI)
%o A242805 { default(primelimit, 1000M); i=0; j=0; k=0; l=0; m=0; loc=0; lb=2; ub=9*10^9; o=3; 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 A242805 (PARI)
%o A242805 is(n) = {my(f=factor(n)); matsize(f)==[3, 2] && vecmax(f[ , 2])==1};
%o A242805 isok(v) = vecextract(v, "^4")==[1, 1, 1, 1, 1, 1]; v = vector(7); for(k=8, 550000, v=concat(vecextract(v, "^1"), is(k+6)); if(isok(v), print1(k, ", "))) \\ _Amiram Eldar_, Nov 13 2018
%o A242805 (PARI) upto(n) = {my(res = List(), streak = 1); for(i = 31, n+6, if(factor(i)[, 2] == [1, 1, 1]~, streak++; if(streak % 4 == 3 && streak >= 7, listput(res, i-6)), if(streak % 4 == 3, streak++, streak = 0))); res} \\ _David A. Corneth_, Nov 13 2018
%Y A242805 Cf. A242793 and A242804 (two primes), A242806 (four primes), A242829 (five primes).
%K A242805 nonn
%O A242805 1,1
%A A242805 _Daniel Constantin Mayer_, May 23 2014