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A109636 Let T(n,k) be the n-th k-almost prime. Then a(n) = T(n,k) such that k is minimal and for all m>0, T(n,k+m) >= 2^m * T(n,k).

%I A109636 #34 Mar 31 2025 02:08:14
%S A109636 2,3,9,10,27,28,30,81,84,88,90,100,104,243,252,264,270,272,280,300,
%T A109636 304,312,729,736,756,784,792,810,816,840,880,900,912,928,936,992,1000,
%U A109636 1040,2187,2208,2268,2352,2368,2376,2430,2448,2464,2520,2624
%N A109636 Let T(n,k) be the n-th k-almost prime. Then a(n) = T(n,k) such that k is minimal and for all m>0, T(n,k+m) >= 2^m * T(n,k).
%C A109636 If one writes the k-almost primes in rows (one row for each k), one observes that there exists a P_{k_0}(n) such that P_{k_0+1}(n) = 2P_{k_0}(n) and for each k>=k_0, P_{k+1}(n)=2P_{k}(n). Then a(n) = P_{k_0}(n). In other words in the columns the values double from row k_0 on. - Peter Pein (petsie(AT)dordos.net), Mar 16 2007
%H A109636 Chai Wah Wu, <a href="/A109636/b109636.txt">Table of n, a(n) for n = 1..10000</a>
%H A109636 Wikipedia, <a href="http://en.wikipedia.org/wiki/Almost_prime">k-almost prime numbers</a>.
%t A109636 a[n_] := Module[{p = Prime[Range[n]], pal}, pal = Transpose /@ Partition[NestList[Take[Union[Flatten[Outer[Times, #1, p]]], Length[#1]] &, p, n], 2, 1]; Complement @@ Transpose[Cases[pal, {k_, kk_} /; kk == 2*k, {2}]]] ; a[50] (* Peter Pein, Nov 10 2007 *)
%o A109636 (Python)
%o A109636 from itertools import count
%o A109636 # uses function A078840_T from A078840
%o A109636 def A109636(n):
%o A109636     a = A078840_T(1,n)
%o A109636     for k in count(2):
%o A109636         b = A078840_T(k,n)
%o A109636         if b==(a<<1):
%o A109636             return a
%o A109636         a = b # _Chai Wah Wu_, Mar 30 2025
%Y A109636 Cf. A000040, A001358, A014612, A014613, A014614, A078840.
%K A109636 nonn
%O A109636 1,1
%A A109636 Yury V. Shlapak (shlapak(AT)imp.kiev.ua), Aug 04 2005
%E A109636 Edited by _Max Alekseyev_, Mar 16 2007
%E A109636 More terms from Peter Pein, Mar 16 2007
%E A109636 Definition corrected by _Chai Wah Wu_, Mar 30 2025