A355382 Number of divisors d of n such that bigomega(d) = omega(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Author

Gus Wiseman, Jul 02 2022

Keywords

• nonn

Comments

The statistic omega = A001221 counts distinct prime factors (without multiplicity).

The statistic bigomega = A001222 counts prime factors with multiplicity.

If positive integers are regarded as arrows from the number of prime factors to the number of distinct prime factors, this sequence counts divisible composable pairs. Is there a nice choice of a composition operation making this into an associative category?

Examples

			The set of divisors of 180 satisfying the condition is {12, 18, 20, 30, 45}, so a(180) = 5.
		

Crossrefs

The version with multiplicity is A181591.

For partitions we have A355383, with multiplicity A339006.

The version for compositions is A355384.

Positions of first appearances are A355386.

A000005 counts divisors.

A001221 counts prime indices without multiplicity.

A001222 count prime indices with multiplicity.

A070175 gives representatives for bigomega and omega, triangle A303555.

Cf. A000712, A022811, A056239, A071625, A118914, A133494, A181819, A182850, A323014, A323022, A323023, A355385, A355388.

Programs

Mathematica

Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeNu[n]&]],{n,100}]