A067003 Number of numbers <= n with same number of distinct prime factors as n.

1, 1, 2, 3, 4, 1, 5, 6, 7, 2, 8, 3, 9, 4, 5, 10, 11, 6, 12, 7, 8, 9, 13, 10, 14, 11, 15, 12, 16, 1, 17, 18, 13, 14, 15, 16, 19, 17, 18, 19, 20, 2, 21, 20, 21, 22, 22, 23, 23, 24, 25, 26, 24, 27, 28, 29, 30, 31, 25, 3, 26, 32, 33, 27, 34, 4, 28, 35, 36, 5, 29, 37, 30, 38, 39, 40, 41

Offset

1,3

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Formula

a(A002110(n)) = 1.

Example

a(11)=8 since 2,3,4,5,7,8,9,11 each have one distinct prime factor. a(12)=3 since 6,10,12 each have two distinct prime factors.
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 distinct prime factors as n:
  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  7  8  6   9   10  11  12  14  13  16  15  17  18
           2  3     4  5  7      8   6   9   10  12  11  13  14  16  15
              2     3  4  5      7       8   6   10  9   11  12  13  14
                    2  3  4      5       7       6   8   9   10  11  12
                       2  3      4       5           7   8   6   9   10
                          2      3       4           5   7       8   6
                                 2       3           4   5       7
                                         2           3   4       5
                                                     2   3       4
                                                         2       3
                                                                 2
(End)

Mathematica

Table[Length[Select[Range[n], PrimeNu[#]==PrimeNu[n]&]], {n, 100}] (* Gus Wiseman, Dec 28 2018 *)

Prog

(PARI) a(n) = my(nb = #factor(n)~); sum(k=1, n, #factor(k)~ == nb); \\ Michel Marcus, Jul 13 2019

Crossrefs

Positions of 1's are A002110.

Cf. A001221, A008479, A058933, A067004. Inverse of A000961, A007774, A033992, A033993, A051270 etc.

Cf. A000010, A006049, A061142, A294277, A294278, A302242, A322837, A322841.

Sequence in context: A133108 A055441 A104717 * A277884 A117386 A101174

Adjacent sequences: A067000 A067001 A067002 * A067004 A067005 A067006

Keyword

nonn,look

Author

Henry Bottomley, Dec 21 2001

Status

approved