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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. 6
 213, 143097, 194757, 206133, 273417, 684897, 807657, 1373937, 1391757, 1516533, 1591593, 1610997, 1774797, 1882977, 1891761, 2046453, 2051493, 2163417, 2163957, 2338053, 2359977, 2522517, 2913837, 3108201, 4221753 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS A remarkable gap occurs between the initial two members, and the sequence seems to be rather sparse compared to the related A242805. 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. All terms are congruent to 9 (mod 12). - Zak Seidov, Apr 14 2015 From Robert Israel, Apr 15 2015: (Start) All terms are congruent to 33 (mod 36). Numbers k in A039833 such that k+4 is in A039833. (End) From Robert G. Wilson v, Apr 15 2015: (Start) k is congruent to 33 (mod 36) so one of its factors is 3 and the other is == 11 (mod 12); k+1 is congruent to 34 (mod 36) so one of its factors is 2 and the other is == 17 (mod 18); k+2 is congruent to 35 (mod 36) so its factors are == +-1 (mod 6); k+4 is congruent to 1 (mod 36) so its factors are == +-1 (mod 6); k+5 is congruent to 2 (mod 36) so one of its factors is 2 and the other is == 1 (mod 18); k+6 is congruent to 3 (mod 36) so one of its factors is 3 and the other is == 1 (mod 12). (End). Number of terms < 10^m: 0, 0, 1, 1, 1, 7, 39, 169, 882, 4852, 27479, ...,. - Robert G. Wilson v, Apr 15 2015 Or, numbers k such that k, k+1 and k+2 are terms in A175648. - Zak Seidov, Dec 08 2015 LINKS Zak Seidov and Robert G. Wilson v, Table of n, a(n) for n = 1..10000 FORMULA a(n) = A202319(n) - 1. - Jon Maiga, Jul 10 2021 EXAMPLE 213=3*71, 214=2*107, 215=5*43, 217=7*31, 218=2*109, 219=3*73. MAPLE f:= t -> numtheory:-issqrfree(t) and (numtheory:-bigomega(t) = 2): 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 MATHEMATICA 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 *) PROG (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

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Last modified January 24 19:03 EST 2022. Contains 350565 sequences. (Running on oeis4.)