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URL: https://oeis.org/A138109

⇱ A138109 - OEIS

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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.
8
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
FORMULA
From Michael De Vlieger, Apr 27 2024: (Start)
Let k = a(n); row k of A162306 = {1, p, q, p^2, p*q}, therefore A010846(k) = 5.
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
(Python)
from math import isqrt
from sympy import primepi, primerange
def A138109(n):
def bisection(f, kmin=0, kmax=1):
while f(kmax) > kmax: kmax <<= 1
kmin = kmax >> 1
while kmax-kmin > 1:
kmid = kmax+kmin>>1
if f(kmid) <= kmid:
kmax = kmid
else:
kmin = kmid
return kmax
def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(min(x//p, p**2)) for p in primerange(s+1)))
return bisection(f, n, n) # Chai Wah Wu, Mar 05 2025
CROSSREFS
Subsequence of A251728 and of A006881. A006094 is a proper subset.
Sequence in context: A261078 A063466 A377713 * A377792 A332877 A357325
KEYWORD
nonn
AUTHOR
David S. Newman, May 04 2008
STATUS
approved