A324923 Number of distinct factors in the factorization of n into factors q(i) = prime(i)/i, i > 0.

0, 1, 2, 1, 3, 2, 2, 1, 2, 3, 4, 2, 3, 2, 3, 1, 3, 2, 2, 3, 3, 4, 3, 2, 3, 3, 2, 2, 4, 3, 5, 1, 4, 3, 4, 2, 3, 2, 3, 3, 4, 3, 3, 4, 3, 3, 4, 2, 2, 3, 4, 3, 2, 2, 4, 2, 3, 4, 4, 3, 3, 5, 3, 1, 4, 4, 3, 3, 3, 4, 4, 2, 4, 3, 3, 2, 5, 3, 5, 3, 2, 4, 4, 3, 5, 3, 4, 4, 3, 3, 4, 3, 5, 4, 4, 2, 4, 2, 4, 3, 4, 4, 3, 3, 4, 2, 3, 2

Offset: 1

Author

Gus Wiseman, Mar 20 2019

Keywords

• nonn

Comments

Also the number of distinct proper terminal subtrees of the rooted tree with Matula-Goebel number n. See illustrations in A061773.

Examples

			The factorization 22 = q(1)^2 q(2) q(3) q(5) has four distinct factors, so a(22) = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Index entries for sequences related to Matula-Goebel numbers
• Index entries for sequences related to prime indices in the factorization of n

Crossrefs

One less than A317713.

Cf. A000081, A003963, A061773, A061775, A109082, A109129, A120383, A196050.

Cf. A324850, A324922, A324924, A324925, A324931, A324934, A324935, A324936, A366385, A366386.

Programs

Mathematica

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Length[Union[difac[n]]],{n,100}]

PARI

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A324923(n) = { my(lista = List([]), gpf, i); while(n > 1, gpf=A006530(n); i = primepi(gpf); n /= gpf; n *= i; listput(lista,i)); #Set(lista); }; \\ Antti Karttunen, Oct 23 2023

Formula

a(n) = A317713(n) - 1.

a(n) = A196050(n) - A366386(n). - Antti Karttunen, Oct 23 2023

Extensions

Data section extended up to a(108) by Antti Karttunen, Oct 23 2023