A028422 Number of ways n can be properly factored.

0, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 4, 0, 3, 0, 3, 1, 1, 0, 6, 1, 1, 2, 3, 0, 4, 0, 6, 1, 1, 1, 8, 0, 1, 1, 6, 0, 4, 0, 3, 3, 1, 0, 11, 1, 3, 1, 3, 0, 6, 1, 6, 1, 1, 0, 10, 0, 1, 3, 10, 1, 4, 0, 3, 1, 4, 0, 15, 0, 1, 3, 3, 1, 4, 0, 11, 4, 1, 0, 10, 1, 1, 1, 6, 0, 10

Offset

1,8

Comments

a(n) = A066032(n,n-1) for n > 1; also number of ways to write n as a product with all factors <= n/2. - Reinhard Zumkeller, Oct 01 2012

Links

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

Formula

From M. F. Hasler, May 27 2026: (Start)

a(n) > 0 iff n is composite. a(n) = 1 iff n is a semiprime.

a(p^k) = A000065(k) = A000041(k) - 1 if p is prime, k >= 0.

a(p1 * p2 * ... * pK) = A000110(K) - 1 if p1, ..., pK are K distinct primes.

a(n) = a(A046523(n)) (= a(m) if m has the same prime signature as n, cf. A118914). (End)

Example

12 can be expressed as 2*6, 3*4 or 2*2*3, so a(12) = 3.

Prog

(Haskell)
a028422 1 = 0
a028422 n = (map (last . init) a066032_tabl) !! (n-1)
-- Reinhard Zumkeller, Oct 01 2012
(PARI) A028422(n)=A001055(n)-1 \\ M. F. Hasler, May 27 2026

Crossrefs

Equals A001055(n) - 1.

Cf. A216599, A216600, A216601, A216602.

Cf. A000041, A000065, A000110, A046523, A118914.

Sequence in context: A294904 A055212 A360661 * A380218 A064577 A371729

Adjacent sequences: A028419 A028420 A028421 * A028423 A028424 A028425

Keyword

nonn

Author

Glen Burch (gburch(AT)erols.com)

Status

approved