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A056809 Numbers n such that n, n+1 and n+2 are products of two primes. 28
33, 85, 93, 121, 141, 201, 213, 217, 301, 393, 445, 633, 697, 841, 921, 1041, 1137, 1261, 1345, 1401, 1641, 1761, 1837, 1893, 1941, 1981, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2721, 2733, 3097, 3385, 3601, 3693, 3865, 3901, 3957, 4285 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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 n were even, then so would be n+2. In fact, one of n or n+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

n+1, which is twice a prime, is in A086005. The primes are in A086006. - T. D. Noe, May 31 2006

LINKS

D. W. Wilson, Table of n, a(n) for n = 1..10000

EXAMPLE

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.

MATHEMATICA

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

f[n_]:=Last/@FactorInteger[n]=={1, 1}||Last/@FactorInteger[n]=={2}; Timing[lst={}; Do[If[f[n]&&f[n+1]&&f[n+2], AppendTo[lst, n]], {n, 2, 8!}]; lst] (* Vladimir Joseph Stephan Orlovsky, Feb 25 2010 *)

Flatten[Position[Partition[PrimeOmega[Range[5000]], 3, 1], {2, 2, 2}]] (* Harvey P. Dale, Feb 15 2015 *)

PROG

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

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

(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

CROSSREFS

Intersection of A070552 and A092207.

Cf. A045939, A039833, A086005, A086006.

Sequence in context: A044171 A044552 A045939 * A073251 A005238 A052214

Adjacent sequences:  A056806 A056807 A056808 * A056810 A056811 A056812

KEYWORD

nonn

AUTHOR

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

EXTENSIONS

Edited and extended by Robert G. Wilson v, May 04 2002

STATUS

approved

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Last modified March 29 18:39 EDT 2017. Contains 284273 sequences.