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

1,1

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.

Links

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

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}.

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A043185 A039362 A043965 * A096512 A243542 A290146

Adjacent sequences: A180147 A180148 A180149 * A180151 A180152 A180153

Keyword

easy,nonn

Author

Jonathan Vos Post, Aug 12 2010

Extensions

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

Status

approved