A056809 Numbers k such that k, k+1 and k+2 are products of two primes.

33, 85, 93, 121, 141, 201, 213, 217, 301, 393, 445, 633, 697, 841, 921, 1041, 1137, 1261, 1345, 1401, 1641, 1761, 1837, 1893, 1941, 1981, 2101, 2181, 2217, 2305, 2361, 2433, 2461, 2517, 2641, 2721, 2733, 3097, 3385, 3601, 3693, 3865, 3901, 3957, 4285

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), May 04 2002

Keywords

• nonn

Comments

Each term is the beginning of a run of three 2-almost primes (semiprimes). No runs exist of length greater than three. For the same reason, each term must be odd: If k were even, then so would be k+2. In fact, one of k or k+2 would be divisible by 4, so must indeed be 4 to have only two prime factors. However, neither 2,3,4 nor 4,5,6 is such a run. - Rick L. Shepherd, May 27 2002

k+1, which is twice a prime, is in A086005. The primes are in A086006. - T. D. Noe, May 31 2006

The squarefree terms are listed in A039833. - Jianing Song, Nov 30 2021

Examples

			121 is in the sequence because 121 = 11^2, 122 = 2*61 and 123 = 3*41, each of which is the product of two primes.
		

Links

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

Crossrefs

Intersection of A070552 and A092207.

Cf. A045939, A039833, A086005, A086006, A123255.

Programs

Mathematica

f[n_] := Plus @@ Transpose[ FactorInteger[n]] [[2]]; Select[Range[10^4], f[ # ] == f[ # + 1] == f[ # + 2] == 2 & ]
Flatten[Position[Partition[PrimeOmega[Range[5000]],3,1],{2,2,2}]] (* Harvey P. Dale, Feb 15 2015 *)
SequencePosition[PrimeOmega[Range[5000]],{2,2,2}][[;;,1]] (* Harvey P. Dale, Mar 03 2024 *)

PARI

forstep(n=1,5000,2, if(bigomega(n)==2 && bigomega(n+1)==2 && bigomega(n+2)==2, print1(n,",")))

PARI

is(n)=n%4==1 && isprime((n+1)/2) && bigomega(n)==2 && bigomega(n+2)==2 \\ Charles R Greathouse IV, Sep 08 2015

PARI

list(lim)=my(v=List(),t); forprime(p=2,(lim+1)\2, if(bigomega(t=2*p-1)==2 && bigomega(t+2)==2, listput(v,t))); Vec(v) \\ Charles R Greathouse IV, Sep 08 2015

Formula

a(n) = A086005(n) - 1 = 2*A086006(n) - 1 = 4*A123255(n) + 1. - Jianing Song, Nov 30 2021

Extensions

Edited and extended by Robert G. Wilson v, May 04 2002