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

6, 12, 15, 18, 24, 35, 36, 45, 48, 54, 72, 75, 77, 96, 108, 135, 143, 144, 162, 175, 192, 216, 221, 225, 245, 288, 323, 324, 375, 384, 405, 432, 437, 486, 539, 576, 648, 667, 675, 768, 847, 864, 875, 899, 972, 1125, 1147, 1152, 1215, 1225, 1296, 1458, 1517, 1536, 1573, 1715, 1728, 1763, 1859, 1875, 1944

Offset: 1

Author

Reinhard Zumkeller, Apr 05 2015

Keywords

• nonn

Examples

			.   n | a(n)                      n | a(n)
. ----+------------------       ----+------------------
.   1 |   6 = 2 * 3              13 |  77 = 7 * 11
.   2 |  12 = 2^2 * 3            14 |  96 = 2^5 * 3
.   3 |  15 = 3 * 5              15 | 108 = 2^2 * 3^3
.   4 |  18 = 2 * 3^2            16 | 135 = 3^3 * 5
.   5 |  24 = 2^3 * 3            17 | 143 = 11 * 13
.   6 |  35 = 5 * 7              18 | 144 = 2^4 * 3^2
.   7 |  36 = 2^2 * 3^2          19 | 162 = 2 * 3^4
.   8 |  45 = 3^2 * 5            20 | 175 = 5^2 * 7
.   9 |  48 = 2^4 * 3            21 | 192 = 2^6 * 3
.  10 |  54 = 2 * 3^3            22 | 216 = 2^3 * 3^3
.  11 |  72 = 2^3 * 3^2          23 | 221 = 13 * 17
.  12 |  75 = 3 * 5^2            24 | 225 = 3^2 * 5^2 .
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A007774.

Subsequences: A006094, A033845, A033849, A033851.

Cf. A000040, A001222, A020639, A006530, A049084, A083553, A151800.

Programs

Haskell

import Data.Set (singleton, deleteFindMin, insert)
a256617 n = a256617_list !! (n-1)
a256617_list = f (singleton (6, 2, 3)) $ tail a000040_list where
   f s ps@(p : ps'@(p':_))
     | m < p * p' = m : f (insert (m * q, q, q')
                          (insert (m * q', q, q') s')) ps
     | otherwise  = f (insert (p * p', p, p') s) ps'
     where ((m, q, q'), s') = deleteFindMin s

Mathematica

Select[Range[2000], MatchQ[FactorInteger[#], {{p_, }, {q, }} /; q == NextPrime[p]]&] (* _Jean-François Alcover, Dec 31 2017 *)

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

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

Python

from sympy import primefactors, nextprime
A256617_list = []
for n in range(1,10**5):
    plist = primefactors(n)
    if len(plist) == 2 and plist[1] == nextprime(plist[0]):
        A256617_list.append(n) # Chai Wah Wu, Aug 23 2021

Formula

A001222(a(n)) = 2.

A006530(a(n)) = A151800(A020639(n)) = A000040(A049084(A020639(a(n)))+1).

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