cp's OEIS Frontend

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.

%I A138109 #31 Mar 05 2025 10:46:50
%S A138109 6,15,21,35,55,65,77,85,91,95,115,119,133,143,161,187,203,209,217,221,
%T A138109 247,253,259,287,299,301,319,323,329,341,377,391,403,407,437,451,473,
%U A138109 481,493,517,527,533,551,559,583,589,611,629,649,667,671,689,697,703
%N 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.
%C A138109 This sequence was suggested by _Moshe Shmuel Newman_.
%C A138109 A020639(n)^2 < a(n) < A020639(n)^3. - _Reinhard Zumkeller_, Dec 17 2014
%C A138109 In other words, k = p*q with primes p, q satisfying p < q < p^2. - _Charles R Greathouse IV_, Apr 03 2017
%C A138109 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
%H A138109 Reinhard Zumkeller, <a href="/A138109/b138109.txt">Table of n, a(n) for n = 1..10000</a>
%F A138109 From _Michael De Vlieger_, Apr 27 2024: (Start)
%F A138109 Let k = a(n); row k of A162306 = {1, p, q, p^2, p*q}, therefore A010846(k) = 5.
%F A138109 A079047(n) = card({ q : p < q < p^2 }), p and q primes. (End)
%e A138109 6 is a term because the smallest prime factor of 6 is 2 and 6^(1/3) = 1.817... < 2 < 2.449... = sqrt(6).
%e A138109 From _Michael De Vlieger_, Apr 27 2024: (Start):
%e A138109 Table of p*q where p = prime(n) and q = prime(n+k):
%e A138109 n\k   1     2     3     4     5     6     7     8     9    10    11
%e A138109 -------------------------------------------------------------------
%e A138109 1:    6;
%e A138109 2:   15,   21;
%e A138109 3:   35,   55,   65,   85,   95,  115;
%e A138109 4:   77,   91,  119,  133,  161,  203,  217,  259,  287,  301,  329;
%e A138109      ... (End)
%t A138109 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]
%t A138109 Select[Range[1000],Surd[#,3]<FactorInteger[#][[1,1]]<Sqrt[#]&] (* _Harvey P. Dale_, May 10 2015 *)
%o A138109 (Haskell)
%o A138109 a138109 n = a138109_list !! (n-1)
%o A138109 a138109_list = filter f [1..] where
%o A138109    f x = p ^ 2 < x && x < p ^ 3 where p = a020639 x
%o A138109 -- _Reinhard Zumkeller_, Dec 17 2014
%o A138109 (PARI) is(n)=my(f=factor(n)); f[,2]==[1,1]~ && f[1,1]^3 > n \\ _Charles R Greathouse IV_, Mar 28 2017
%o A138109 (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
%o A138109 (Python)
%o A138109 from math import isqrt
%o A138109 from sympy import primepi, primerange
%o A138109 def A138109(n):
%o A138109     def bisection(f,kmin=0,kmax=1):
%o A138109         while f(kmax) > kmax: kmax <<= 1
%o A138109         kmin = kmax >> 1
%o A138109         while kmax-kmin > 1:
%o A138109             kmid = kmax+kmin>>1
%o A138109             if f(kmid) <= kmid:
%o A138109                 kmax = kmid
%o A138109             else:
%o A138109                 kmin = kmid
%o A138109         return kmax
%o A138109     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)))
%o A138109     return bisection(f,n,n) # _Chai Wah Wu_, Mar 05 2025
%Y A138109 Subsequence of A251728 and of A006881. A006094 is a proper subset.
%Y A138109 Cf. A020639, A010846, A079047, A162306.
%K A138109 nonn
%O A138109 1,1
%A A138109 _David S. Newman_, May 04 2008