OFFSET
1,1
COMMENTS
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).
A subset of A045940. - Zak Seidov, Nov 05 2006
LINKS
D. W. Wilson, Table of n, a(n) for n = 1..10000
EXAMPLE
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.
MAPLE
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
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Nov 03 2006
EXTENSIONS
More terms from Zak Seidov, Nov 05 2006
More terms from Emeric Deutsch, Nov 07 2006
STATUS
approved