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A058933 a(1)=1; for k > 0, if n is k-almost prime then a(n) = number of k-almost primes <= n. 10
1, 1, 2, 1, 3, 2, 4, 1, 3, 4, 5, 2, 6, 5, 6, 1, 7, 3, 8, 4, 7, 8, 9, 2, 9, 10, 5, 6, 10, 7, 11, 1, 11, 12, 13, 3, 12, 14, 15, 4, 13, 8, 14, 9, 10, 16, 15, 2, 17, 11, 18, 12, 16, 5, 19, 6, 20, 21, 17, 7, 18, 22, 13, 1, 23, 14, 19, 15, 24, 16, 20, 3, 21, 25, 17, 18, 26, 19, 22, 4, 8, 27, 23 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Also the number of positive integers less than or equal to n with the same number of prime factors as n, counted with multiplicity. - Gus Wiseman, Dec 28 2018

There is a close relationship between a(n) and a(n^2). See A209934 for an exploratory quantification. - Peter Munn, Aug 04 2019

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..20000

FORMULA

Ordinal transform of A001222 (bigomega). - Franklin T. Adams-Watters, Aug 28 2006

If a(n) < a(3^A001222(2n)) = A078843(A001222(2n)) then a(2n) = a(n), otherwise a(2n) > a(n). - Peter Munn, Aug 05 2019

EXAMPLE

3 is prime, so a(3)=2. 10 is 2-almost prime (semiprime), so a(10)=4.

From Gus Wiseman, Dec 28 2018: (Start)

Column n lists the a(n) positive integers less than or equal to n with the same number of prime factors as n, counted with multiplicity:

  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20

  ---------------------------------------------------------------------

  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20

        2     3  4  5     6  9   7   8   11  10  14      13  12  17  18

              2     3     4  6   5       7   9   10      11  8   13  12

                    2        4   3       5   6   9       7       11  8

                                 2       3   4   6       5       7

                                         2       4       3       5

                                                         2       3

                                                                 2

(End)

MAPLE

p:= proc() 0 end:

a:= proc(n) option remember; local t;

      t:= numtheory[bigomega](n);

      p(t):= p(t)+1

    end:

seq(a(n), n=1..100);  # Alois P. Heinz, Oct 09 2015

MATHEMATICA

p[_] = 0; a[n_] := a[n] = Module[{t}, t = PrimeOmega[n]; p[t] = p[t]+1]; Table[a[n], {n, 1, 100}] (* Jean-Fran├žois Alcover, Jan 24 2017, after Alois P. Heinz *)

CROSSREFS

Positions of 1's are A000079.

Cf. A001358, A014612, A014613, A014614.

Cf. A000010, A000961, A001222, A006049, A045920, A061142, A067003, A209934, A302242, A322838, A322839, A322840.

Sequence in context: A124172 A336394 A336472 * A087470 A191475 A158456

Adjacent sequences:  A058930 A058931 A058932 * A058934 A058935 A058936

KEYWORD

easy,nonn

AUTHOR

Naohiro Nomoto, Jan 11 2001

STATUS

approved

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Last modified November 26 12:37 EST 2020. Contains 338639 sequences. (Running on oeis4.)