OFFSET
1,1
COMMENTS
This sequence was suggested by Moshe Shmuel Newman.
In other words, k = p*q with primes p, q satisfying p < q < p^2. - Charles R Greathouse IV, Apr 03 2017
If "strictly less than" in the definition were changed to "less than or equal to" then this sequence would also include the squares of primes (A001248), resulting in A251728. - Jon E. Schoenfield, Dec 27 2022
LINKS
Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
FORMULA
From Michael De Vlieger, Apr 27 2024: (Start)
A079047(n) = card({ q : p < q < p^2 }), p and q primes. (End)
EXAMPLE
6 is a term because the smallest prime factor of 6 is 2 and 6^(1/3) = 1.817... < 2 < 2.449... = sqrt(6).
From Michael De Vlieger, Apr 27 2024: (Start):
Table of p*q where p = prime(n) and q = prime(n+k):
n\k 1 2 3 4 5 6 7 8 9 10 11
-------------------------------------------------------------------
1: 6;
2: 15, 21;
3: 35, 55, 65, 85, 95, 115;
4: 77, 91, 119, 133, 161, 203, 217, 259, 287, 301, 329;
... (End)
MATHEMATICA
s = {}; Do[f = FactorInteger[i]; test = f[[1]][[1]]; If [test < N[i^(1/2)] && test > N[i^(1/3)], s = Union[s, {i}]], {i, 2, 2000}]; Print[s]
Select[Range[1000], Surd[#, 3]<FactorInteger[#][[1, 1]]<Sqrt[#]&] (* Harvey P. Dale, May 10 2015 *)
PROG
(Haskell)
a138109 n = a138109_list !! (n-1)
a138109_list = filter f [1..] where
f x = p ^ 2 < x && x < p ^ 3 where p = a020639 x
-- Reinhard Zumkeller, Dec 17 2014
(PARI) is(n)=my(f=factor(n)); f[, 2]==[1, 1]~ && f[1, 1]^3 > n \\ Charles R Greathouse IV, Mar 28 2017
(PARI) list(lim)=if(lim<6, return([])); my(v=List([6])); forprime(p=3, sqrtint(1+lim\=1)-1, forprime(q=p+2, min(p^2-2, lim\p), listput(v, p*q))); Set(v) \\ Charles R Greathouse IV, Mar 28 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
David S. Newman, May 04 2008
STATUS
approved