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

1,3

Comments

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

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

Formula

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

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

Example

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

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}]

Prog

(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

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.

Sequence in context: A303639 A090000 A109082 * A126303 A306467 A157810

Adjacent sequences: A324920 A324921 A324922 * A324924 A324925 A324926

Keyword

nonn

Author

Gus Wiseman, Mar 20 2019

Extensions

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

Status

approved