A124057 Numbers n such that n, n+1, n+2 and n+3 are products of exactly 3 primes.

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

Offset: 1

Author

Jonathan Vos Post, Nov 03 2006

Keywords

• easy
• nonn

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

Examples

			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.
		

Links

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

Crossrefs

Cf. A000040, A014612, A056809, A067813, A113789, A045940.

Programs

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 *)

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

Formula

n, n+1, n+2 and n+3 are all elements of A014612. n and n+1 are elements of A113789.

Extensions

More terms from Zak Seidov, Nov 05 2006

More terms from Emeric Deutsch, Nov 07 2006