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

1,1

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A352198 A003140 A057234 * A057236 A063257 A229492

Adjacent sequences: A109633 A109634 A109635 * A109637 A109638 A109639

Keyword

nonn

Author

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

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

Status

approved