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).

2, 3, 9, 10, 27, 28, 30, 81, 84, 88, 90, 100, 104, 243, 252, 264, 270, 272, 280, 300, 304, 312, 729, 736, 756, 784, 792, 810, 816, 840, 880, 900, 912, 928, 936, 992, 1000, 1040, 2187, 2208, 2268, 2352, 2368, 2376, 2430, 2448, 2464, 2520, 2624

Offset: 1

Author

Yury V. Shlapak (shlapak(AT)imp.kiev.ua), Aug 04 2005

Keywords

• nonn

Comments

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

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000
• Wikipedia, k-almost prime numbers.

Crossrefs

Cf. A000040, A001358, A014612, A014613, A014614, A078840.

Programs

Mathematica

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 *)

Python

from itertools import count
# uses function A078840_T from A078840
def A109636(n):
    a = A078840_T(1,n)
    for k in count(2):
        b = A078840_T(k,n)
        if b==(a<<1):
            return a
        a = b # Chai Wah Wu, Mar 30 2025

Extensions

Edited by Max Alekseyev, Mar 16 2007

More terms from Peter Pein, Mar 16 2007

Definition corrected by Chai Wah Wu, Mar 30 2025