A113789 Numbers n such that n, n+1 and n+2 are products of exactly 3 primes.

170, 244, 284, 428, 434, 506, 602, 603, 604, 637, 962, 1074, 1083, 1084, 1130, 1244, 1309, 1412, 1434, 1490, 1532, 1556, 1586, 1604, 1634, 1675, 1771, 1885, 1946, 2012, 2013, 2035, 2084, 2091, 2092, 2162, 2396, 2404, 2522, 2523, 2524, 2525, 2634, 2635

Offset: 1

Author

Jonathan Vos Post, Jan 21 2006

Keywords

• easy
• nonn

Comments

3-almost prime analog of A056809.

This sequence consists of the least of 3 consecutive 3-almost primes, or 4 or more consecutive 3-almost primes (i.e. n, n+1 and n+2 but not excluding n+3 also 3-almost prime). 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).

Primes counted with multiplicity. - Harvey P. Dale, Sep 04 2019

Examples

			a(1) = 170 because 170 = 2 * 5 * 17 and 171 = 3^2 * 19 and 172 = 2^2 * 43 are all 3-almost primes.
a(2) = 244 because 244 = 2^2 * 61 and 245 = 5 * 7^2 and 246 = 2 * 3 * 41 are all 3-almost primes.
a(3) = 284 because 284 = 2^2 * 71 and 285 = 3 * 5 * 19 and 286 = 2 * 11 * 13 are all 3-almost primes.
a(4) = 428 because 428 = 2^2 * 107 and 429 = 3 * 11 * 13 and 430 = 2 * 5 * 43 are all 3-almost primes.
a(5) = 434 because 434 = 2 * 7 * 31 and 435 = 3 * 5 * 29 and 436 = 2^2 * 109 are all 3-almost primes.
a(6) = 506 because 506 = 2 * 11 * 23 and 507 = 3 * 13^2 and 508 = 2^2 * 127 all 3-almost primes.
a(7), a(8), a(9) = 602, 603, 604 because of the record-setting 5 consecutive 3-almost primes: 602 = 2 * 7 * 43; 603 = 3^2 * 67; 604 = 2^2 * 151; 605 = 5 * 11^2; 606 = 2 * 3 * 101.
		

Links

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

Crossrefs

Cf. A014612, A056809, A067813.

Subsequence of A180117.

Programs

Mathematica

fQ[n_] := Plus @@ Last /@ FactorInteger@n == 3; Select[ Range@2664, fQ@# && fQ[ # + 1] && fQ[ # + 2] &] (* Robert G. Wilson v, Jan 21 2006 *)
SequencePosition[Table[If[PrimeOmega[n]==3,1,0],{n,3000}],{1,1,1}][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 04 2019 *)

PARI

is(n)=bigomega(n)==3 && bigomega(n+1)==3 && bigomega(n+2)==3 \\ Charles R Greathouse IV, Feb 05 2017

Formula

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

Extensions

Edited, corrected and extended by Robert G. Wilson v, Jan 21 2006