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A124057
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Numbers n such that n, n+1, n+2 and n+3 are products of exactly 3 primes.
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4
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602, 603, 1083, 2012, 2091, 2522, 2523, 2524, 2634, 2763, 3243, 3355, 4202, 4203, 4921, 4922, 4923, 5034, 5035, 5132, 5203, 5282, 5283, 5785, 5882, 5954, 5972, 6092, 6212, 6476, 6962, 6985, 7730, 7731, 7945, 8393, 8825, 8956, 8972, 9188, 9482, 10011
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OFFSET
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1,1
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COMMENTS
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n such that n, n+1, n+2 and n+3 are 3-almost primes. Subset of A113789 Numbers n such that n, n+1 and n+2 are products of exactly 3 primes. A067813 has some runs of up to 7 consecutive 3-almost primes (i.e. starting 211673). But there cannot be 8 consecutive 3-almost primes, as every run of 8 consecutive positive integers contains exactly one multiple of 8 = 2^3 and only 8 of all positive multiples of 8 is a 3-almost prime (i.e. all larger multiples have at least 4 prime factors, with multiplicity).
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LINKS
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FORMULA
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n, n+1, n+2 and n+3 are all elements of A014612. n and n+1 are elements of A113789.
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EXAMPLE
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a(1) = 602 because 602 = 2 * 7 * 43 and 603 = 3^2 * 67 and 604 = 2^2 * 151 and 605 = 5 * 11^2 are all 3-almost primes.
a(2) = 603 because 603 = 3^2 * 67 and 604 = 2^2 * 151 and 605 = 5 * 11^2 and 606 = 2 * 3 * 101 are all 3-almost primes.
a(3) = 1083 because 1083 = 3 * 19^2 and 1084 = 2^2 * 271 and 1085 = 5 * 7 * 31 and 1086 = 2 * 3 * 181 are all 3-almost primes.
a(4) = 2012 because 2012 = 2^2 * 503, 2013 = 3 * 11 * 61, 2014 = 2 * 19 * 53, 2015 = 5 * 13 * 31.
a(5) = 2091 because 2091 = 3 * 17 * 41, 2092 = 2^2 * 523, 2093 = 7 * 13 * 23, 2094 = 2 * 3 * 349.
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MAPLE
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with(numtheory): a:=proc(n) if bigomega(n)=3 and bigomega(n+1)=3 and bigomega(n+2)=3 and bigomega(n+3)=3 then n else fi end: seq(a(n), n=1..15000); # Emeric Deutsch, Nov 07 2006
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MATHEMATICA
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okQ[{a_, b_, c_, d_}]:=Union[{a, b, c, d}]=={3}; Flatten[Position[Partition[ PrimeOmega[ Range[11000]], 4, 1], _?(okQ)]] (* Harvey P. Dale, Sep 23 2012 *)
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PROG
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(PARI) is(n)=if(!isprime((n+3)\4), return(0)); for(k=n, n+3, if(bigomega(k)!=3, return(0))); 1 \\ Charles R Greathouse IV, Feb 05 2017
(PARI) list(lim)=my(v=List(), u=v, t); forprime(p=2, lim\4, forprime(q=2, min(lim\(2*p), p), t=p*q; forprime(r=2, min(lim\t, q), listput(u, t*r)))); u=Set(u); for(i=4, #u, if(u[i]-u[i-3]==3, listput(v, u[i-3]))); Vec(v) \\ Charles R Greathouse IV, Feb 05 2017
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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