A327486 Product of Omegas of positive integers from 2 to n.

1, 1, 1, 2, 2, 4, 4, 12, 24, 48, 48, 144, 144, 288, 576, 2304, 2304, 6912, 6912, 20736, 41472, 82944, 82944, 331776, 663552, 1327104, 3981312, 11943936, 11943936, 35831808, 35831808, 179159040, 358318080, 716636160, 1433272320, 5733089280, 5733089280, 11466178560

Offset: 1

Author

Gus Wiseman, Sep 28 2019

Keywords

• nonn

Comments

Omega(n) (A001222) is the number of prime factors of n, counted with multiplicity.

Also the number of ways to choose a prime factor, counting multiplicity, of each positive integer from 2 to n.

Examples

			We have a(8) = 1 * 1 * 2 * 1 * 2 * 1 * 3 = 12.
		

Crossrefs

Partial products of A001222.

Cf. A001221, A056239, A071625, A112798, A118914, A316413, A327485.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, n,
      numtheory[bigomega](n)*a(n-1))
    end:
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 29 2019

Mathematica

Table[Product[PrimeOmega[k],{k,2,n}],{n,30}]

PARI

a(n) = prod(k=2, n, bigomega(k)); \\ Michel Marcus, Sep 29 2019

Formula

a(n > 1) = a(n - 1) * A001222(n).