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
Examples
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)
Links
- Rémy Sigrist, Table of n, a(n) for n = 1..10000
Crossrefs
Programs
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Mathematica
Table[Length[Select[Range[n],PrimeNu[#]==PrimeNu[n]&]],{n,100}] (* Gus Wiseman, Dec 28 2018 *) -
PARI
a(n) = my(nb = #factor(n)~); sum(k=1, n, #factor(k)~ == nb); \\ Michel Marcus, Jul 13 2019
Formula
a(A002110(n)) = 1.