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%I #23 Dec 28 2022 02:14:07
%S 6,15,21,35,55,65,77,85,91,95,115,119,133,143,161,187,203,209,217,221,
%T 247,253,259,287,299,301,319,323,329,341,377,391,403,407,437,451,473,
%U 481,493,517,527,533,551,559,583,589,611,629,649,667,671,689,697,703
%N 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 This sequence was suggested by _Moshe Shmuel Newman_.
%C A020639(n)^2 < a(n) < A020639(n)^3. - _Reinhard Zumkeller_, Dec 17 2014
%C In other words, k = p*q with primes p, q satisfying p < q < p^2. - _Charles R Greathouse IV_, Apr 03 2017
%C 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 Reinhard Zumkeller, <a href="/A138109/b138109.txt">Table of n, a(n) for n = 1..10000</a>
%e 6 is a term because the smallest prime factor of 6 is 2 and 6^(1/3) = 1.817... < 2 < 2.449... = sqrt(6).
%t 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 Select[Range[1000],Surd[#,3]<FactorInteger[#][[1,1]]<Sqrt[#]&] (* _Harvey P. Dale_, May 10 2015 *)
%o (Haskell)
%o a138109 n = a138109_list !! (n-1)
%o a138109_list = filter f [1..] where
%o f x = p ^ 2 < x && x < p ^ 3 where p = a020639 x
%o -- _Reinhard Zumkeller_, Dec 17 2014
%o (PARI) is(n)=my(f=factor(n)); f[,2]==[1,1]~ && f[1,1]^3 > n \\ _Charles R Greathouse IV_, Mar 28 2017
%o (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
%Y Subsequence of A251728 and of A006881.
%Y Cf. A020639.
%K nonn
%O 1,1
%A _David S. Newman_, May 04 2008
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