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

1,2

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).

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Prime Factorization.

Formula

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000961, A006881, A008480, A099849.

Sequence in context: A291764 A319681 A340767 * A082287 A331267 A238839

Adjacent sequences: A099845 A099846 A099847 * A099849 A099850 A099851

Keyword

nonn,easy

Author

Reinhard Zumkeller, Oct 27 2004

Status

approved