A123255 -id:A123255 - cp's OEIS frontend

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

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

nonn

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

			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.

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

Intersection of A070552 and A092207.

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

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 *)

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

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

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

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

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

A086005 Semiprimes sandwiched between semiprimes.

Original entry on oeis.org

34, 86, 94, 122, 142, 202, 214, 218, 302, 394, 446, 634, 698, 842, 922, 1042, 1138, 1262, 1346, 1402, 1642, 1762, 1838, 1894, 1942, 1982, 2102, 2182, 2218, 2306, 2362, 2434, 2462, 2518, 2642, 2722, 2734, 3098, 3386, 3602, 3694, 3866, 3902, 3958, 4286, 4414

Offset: 1

Reinhard Zumkeller, Jul 07 2003

nonn

These are some of the balanced semiprimes (see A213025). - Alonso del Arte, Jun 04 2012

			94 = 47*2: 94 - 1 = 3*31 and 94 + 1 = 5*19, therefore 94 is in the sequence.

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Semiprime

Cf. A001358, A056809, A064911, A100484, A123255.

a086005 n = a086005_list !! (n-1)
a086005_list = filter
   (\x -> a064911 (x - 1) == 1 && a064911 (x + 1) == 1) a100484_list
-- Reinhard Zumkeller, Aug 08 2013, Jun 10 2012

u[n_]:=Plus@@Last/@FactorInteger[n]==2;lst={};Do[If[u[n],sp=n;If[u[sp-1]&&u[sp+1],AppendTo[lst,sp]]],{n,8!}];lst  (* Vladimir Joseph Stephan Orlovsky, Nov 16 2009 *)
(* First run program for A109611 to define semiPrimeQ *) Select[Range[4000], Union[{semiPrimeQ[# - 1], semiPrimeQ[#], semiPrimeQ[# + 1]}] == {True} &] (* Alonso del Arte, Jun 03 2012 *)
Select[Partition[Range@ 4000, 3, 1], Union@ PrimeOmega@ # == {2} &][[All, 2]] (* Michael De Vlieger, Jun 14 2017 *)

from itertools import count, islice
from sympy import factorint, isprime
def agen(): # generator of terms
    nxt = 0
    for k in count(2, 2):
        prv, nxt = nxt, sum(factorint(k+1).values())
        if prv == nxt == 2 and isprime(k//2): yield k
print(list(islice(agen(), 46))) # Michael S. Branicky, Nov 26 2022

a(n) = 2*A086006(n).

a(n) = A056809(n)+1. - Zak Seidov, Sep 30 2012

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

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A086005 Semiprimes sandwiched between semiprimes.

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