%I #55 Mar 03 2024 16:02:44

%S 33,85,93,121,141,201,213,217,301,393,445,633,697,841,921,1041,1137,

%T 1261,1345,1401,1641,1761,1837,1893,1941,1981,2101,2181,2217,2305,

%U 2361,2433,2461,2517,2641,2721,2733,3097,3385,3601,3693,3865,3901,3957,4285

%N Numbers k such that k, k+1 and k+2 are products of two primes.

%C Each term is the beginning of a run of three 2-almost primes (semiprimes). No runs exist of length greater than three. For the same reason, each term must be odd: If k were even, then so would be k+2. In fact, one of k or k+2 would be divisible by 4, so must indeed be 4 to have only two prime factors. However, neither 2,3,4 nor 4,5,6 is such a run. - _Rick L. Shepherd_, May 27 2002

%C k+1, which is twice a prime, is in A086005. The primes are in A086006. - _T. D. Noe_, May 31 2006

%C The squarefree terms are listed in A039833. - _Jianing Song_, Nov 30 2021

%H D. W. Wilson, <a href="/A056809/b056809.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A086005(n) - 1 = 2*A086006(n) - 1 = 4*A123255(n) + 1. - _Jianing Song_, Nov 30 2021

%e 121 is in the sequence because 121 = 11^2, 122 = 2*61 and 123 = 3*41, each of which is the product of two primes.

%t f[n_] := Plus @@ Transpose[ FactorInteger[n]] [[2]]; Select[Range[10^4], f[ # ] == f[ # + 1] == f[ # + 2] == 2 & ]

%t Flatten[Position[Partition[PrimeOmega[Range[5000]],3,1],{2,2,2}]] (* _Harvey P. Dale_, Feb 15 2015 *)

%t SequencePosition[PrimeOmega[Range[5000]],{2,2,2}][[;;,1]] (* _Harvey P. Dale_, Mar 03 2024 *)

%o (PARI) forstep(n=1,5000,2, if(bigomega(n)==2 && bigomega(n+1)==2 && bigomega(n+2)==2, print1(n,",")))

%o (PARI) is(n)=n%4==1 && isprime((n+1)/2) && bigomega(n)==2 && bigomega(n+2)==2 \\ _Charles R Greathouse IV_, Sep 08 2015

%o (PARI) list(lim)=my(v=List(),t); forprime(p=2,(lim+1)\2, if(bigomega(t=2*p-1)==2 && bigomega(t+2)==2, listput(v,t))); Vec(v) \\ _Charles R Greathouse IV_, Sep 08 2015

%Y Intersection of A070552 and A092207.

%Y Cf. A045939, A039833, A086005, A086006, A123255.

%K nonn

%O 1,1

%A Sharon Sela (sharonsela(AT)hotmail.com), May 04 2002

%E Edited and extended by _Robert G. Wilson v_, May 04 2002