A180150 Numbers n such that n and n+2 are both divisible by exactly 4 primes (counted with multiplicity).

54, 88, 150, 196, 232, 248, 294, 306, 328, 340, 342, 348, 460, 488, 490, 568, 570, 664, 712, 738, 774, 850, 856, 858, 868, 870, 948, 1012, 1014, 1060, 1096, 1110, 1148, 1190, 1204, 1206, 1208, 1210, 1218, 1254, 1274, 1276, 1290, 1302, 1314, 1420, 1430, 1448

Offset: 1

Author

Jonathan Vos Post, Aug 12 2010

Keywords

• easy
• nonn

Comments

"Quadruprimes" or "4-almost primes" that keep that property when incremented by 2. This sequence is to 4 as 3 is to A180117, as A092207 is to 2, and as A001359 is to 1. That is, this sequence is the 4th row of the infinite array A[k,n] = n-th natural number m such that m and m+2 are both divisible by exactly k primes (counted with multiplicity). The first row is the lesser of twin primes. The second row is the sequence such that m and m+2 are both semiprimes.

Examples

			a(1) = 54 because 54 = 2 * 3^3 is divisible by exactly 4 primes (counted with multiplicity), and so is 54+2 = 56 = 2^3 * 7.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A001222, A001358, A014614, A033987, A101637, A114106 (number of 4-almost primes <= 10^n).

Programs

Mathematica

SequencePosition[PrimeOmega[Range[1500]],{4,,4}][[;;,1]] (* _Harvey P. Dale, Jan 14 2024 *)

PARI

is(n)=bigomega(n)==4 && bigomega(n+2)==4 \\ Charles R Greathouse IV, Jan 31 2017

Formula

{m in A014613 and m+2 in A014613} = {m such that bigomega(m) = bigomega(m+2) = 4} = {m such that A001222(m) = A001222(m+2) = 4}.

Extensions

More terms from R. J. Mathar, Aug 13 2010