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A056809
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Numbers k such that k, k+1 and k+2 are products of two primes.
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31
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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FORMULA
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Sharon Sela (sharonsela(AT)hotmail.com), May 04 2002
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EXTENSIONS
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STATUS
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approved
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