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A056809 Numbers k such that k, k+1 and k+2 are products of two primes. 30
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 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
k+1, which is twice a prime, is in A086005. The primes are in A086006. - T. D. Noe, May 31 2006
The squarefree terms are listed in A039833. - Jianing Song, Nov 30 2021
LINKS
FORMULA
a(n) = A086005(n) - 1 = 2*A086006(n) - 1 = 4*A123255(n) + 1. - Jianing Song, Nov 30 2021
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 & ]
Flatten[Position[Partition[PrimeOmega[Range[5000]], 3, 1], {2, 2, 2}]] (* Harvey P. Dale, Feb 15 2015 *)
SequencePosition[PrimeOmega[Range[5000]], {2, 2, 2}][[;; , 1]] (* Harvey P. Dale, Mar 03 2024 *)
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.
Sequence in context: A044171 A044552 A045939 * A073251 A005238 A052214
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 05:48 EDT 2024. Contains 371265 sequences. (Running on oeis4.)