A138109 Positive integers k whose smallest prime factor is greater than the cube root of k and strictly less than the square root of k.

6, 15, 21, 35, 55, 65, 77, 85, 91, 95, 115, 119, 133, 143, 161, 187, 203, 209, 217, 221, 247, 253, 259, 287, 299, 301, 319, 323, 329, 341, 377, 391, 403, 407, 437, 451, 473, 481, 493, 517, 527, 533, 551, 559, 583, 589, 611, 629, 649, 667, 671, 689, 697, 703

Offset

1,1

Comments

This sequence was suggested by Moshe Shmuel Newman.

A020639(n)^2 < a(n) < A020639(n)^3. - Reinhard Zumkeller, Dec 17 2014

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

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).

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

Subsequence of A251728 and of A006881.

Cf. A020639.

Sequence in context: A015793 A261078 A063466 * A332877 A357325 A362211

Adjacent sequences: A138106 A138107 A138108 * A138110 A138111 A138112

Keyword

nonn

Author

David S. Newman, May 04 2008

Status

approved