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A256617 Numbers having exactly two distinct prime factors, which are also adjacent prime numbers.

%I A256617 #26 Aug 23 2021 00:38:53
%S A256617 6,12,15,18,24,35,36,45,48,54,72,75,77,96,108,135,143,144,162,175,192,
%T A256617 216,221,225,245,288,323,324,375,384,405,432,437,486,539,576,648,667,
%U A256617 675,768,847,864,875,899,972,1125,1147,1152,1215,1225,1296,1458,1517,1536,1573,1715,1728,1763,1859,1875,1944
%N A256617 Numbers having exactly two distinct prime factors, which are also adjacent prime numbers.
%H A256617 Reinhard Zumkeller, <a href="/A256617/b256617.txt">Table of n, a(n) for n = 1..10000</a>
%F A256617 A001222(a(n)) = 2.
%F A256617 A006530(a(n)) = A151800(A020639(n)) = A000040(A049084(A020639(a(n)))+1).
%F A256617 Sum_{n>=1} 1/a(n) = Sum_{n>=1} 1/A083553(n) = Sum_{n>=1} 1/((prime(n)-1)*(prime(n+1)-1)) = 0.7126073495... - _Amiram Eldar_, Dec 23 2020
%e A256617 .   n | a(n)                      n | a(n)
%e A256617 . ----+------------------       ----+------------------
%e A256617 .   1 |   6 = 2 * 3              13 |  77 = 7 * 11
%e A256617 .   2 |  12 = 2^2 * 3            14 |  96 = 2^5 * 3
%e A256617 .   3 |  15 = 3 * 5              15 | 108 = 2^2 * 3^3
%e A256617 .   4 |  18 = 2 * 3^2            16 | 135 = 3^3 * 5
%e A256617 .   5 |  24 = 2^3 * 3            17 | 143 = 11 * 13
%e A256617 .   6 |  35 = 5 * 7              18 | 144 = 2^4 * 3^2
%e A256617 .   7 |  36 = 2^2 * 3^2          19 | 162 = 2 * 3^4
%e A256617 .   8 |  45 = 3^2 * 5            20 | 175 = 5^2 * 7
%e A256617 .   9 |  48 = 2^4 * 3            21 | 192 = 2^6 * 3
%e A256617 .  10 |  54 = 2 * 3^3            22 | 216 = 2^3 * 3^3
%e A256617 .  11 |  72 = 2^3 * 3^2          23 | 221 = 13 * 17
%e A256617 .  12 |  75 = 3 * 5^2            24 | 225 = 3^2 * 5^2 .
%t A256617 Select[Range[2000], MatchQ[FactorInteger[#], {{p_, _}, {q_, _}} /; q == NextPrime[p]]&] (* _Jean-François Alcover_, Dec 31 2017 *)
%o A256617 (Haskell)
%o A256617 import Data.Set (singleton, deleteFindMin, insert)
%o A256617 a256617 n = a256617_list !! (n-1)
%o A256617 a256617_list = f (singleton (6, 2, 3)) $ tail a000040_list where
%o A256617    f s ps@(p : ps'@(p':_))
%o A256617      | m < p * p' = m : f (insert (m * q, q, q')
%o A256617                           (insert (m * q', q, q') s')) ps
%o A256617      | otherwise  = f (insert (p * p', p, p') s) ps'
%o A256617      where ((m, q, q'), s') = deleteFindMin s
%o A256617 (PARI) is(n) = if(omega(n)!=2, return(0), my(f=factor(n)[, 1]~); if(f[2]==nextprime(f[1]+1), return(1))); 0 \\ _Felix Fröhlich_, Dec 31 2017
%o A256617 (PARI) list(lim)=my(v=List(),c=sqrtnint(lim\=1,3),d=nextprime(c+1),p=2); forprime(q=3,d, for(i=1,logint(lim\q,p), my(t=p^i); while((t*=q)<=lim, listput(v,t))); p=q); forprime(q=d+1,lim\precprime(sqrtint(lim)), listput(v,p*q); p=q); Set(v) \\ _Charles R Greathouse IV_, Apr 12 2020
%o A256617 (Python)
%o A256617 from sympy import primefactors, nextprime
%o A256617 A256617_list = []
%o A256617 for n in range(1,10**5):
%o A256617     plist = primefactors(n)
%o A256617     if len(plist) == 2 and plist[1] == nextprime(plist[0]):
%o A256617         A256617_list.append(n) # _Chai Wah Wu_, Aug 23 2021
%Y A256617 Subsequence of A007774.
%Y A256617 Subsequences: A006094, A033845, A033849, A033851.
%Y A256617 Cf. A000040, A001222, A020639, A006530, A049084, A083553, A151800.
%K A256617 nonn
%O A256617 1,1
%A A256617 _Reinhard Zumkeller_, Apr 05 2015