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A256617 Numbers having exactly two distinct prime factors, which are also adjacent prime numbers. 15
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
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
EXAMPLE
. 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 .
MATHEMATICA
Select[Range[2000], MatchQ[FactorInteger[#], {{p_, _}, {q_, _}} /; q == NextPrime[p]]&] (* Jean-François Alcover, Dec 31 2017 *)
PROG
(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
(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
CROSSREFS
Subsequence of A007774.
Subsequences: A006094, A033845, A033849, A033851.
Sequence in context: A309944 A212308 A089341 * A252044 A315617 A287572
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Apr 05 2015
STATUS
approved

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Last modified March 19 04:58 EDT 2024. Contains 370952 sequences. (Running on oeis4.)