A099848 All natural numbers occur in their order as many times as they have ordered prime factorizations.

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

Offset: 1

Author

Reinhard Zumkeller, Oct 27 2004

Keywords

• nonn
• easy

Comments

k occurs exactly once iff k is a prime power (A000961).

k occurs exactly twice iff k is a semiprime with distinct factors (A006881).

Examples

			k = 12: A008480(12) = #{3*2*2, 2*3*2, 2*2*3} = 3, therefore 12 occurs three times.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime Factorization.

Crossrefs

Cf. A000961, A006881, A008480, A099849.

Programs

Haskell

a099848 n = a099848_list !! (n-1)
a099848_list = concat $ zipWith replicate a008480_list [1..]
-- Reinhard Zumkeller, Nov 19 2015

Mathematica

Table[n, {n, 1, 40}, {Multinomial @@ FactorInteger[n][[;; , 2]]}] // Flatten (* Amiram Eldar, May 13 2025 *)

Formula

a(A099849(n) - k) = n for 0 <= k < A008480(n).