cp's OEIS Frontend

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

%I A324923 #13 Oct 23 2023 15:17:24
%S A324923 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,
%T A324923 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,
%U A324923 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
%N A324923 Number of distinct factors in the factorization of n into factors q(i) = prime(i)/i, i > 0.
%C A324923 Also the number of distinct proper terminal subtrees of the rooted tree with Matula-Goebel number n. See illustrations in A061773.
%H A324923 Antti Karttunen, <a href="/A324923/b324923.txt">Table of n, a(n) for n = 1..65537</a>
%H A324923 <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>
%H A324923 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>
%F A324923 a(n) = A317713(n) - 1.
%F A324923 a(n) = A196050(n) - A366386(n). - _Antti Karttunen_, Oct 23 2023
%e A324923 The factorization 22 = q(1)^2 q(2) q(3) q(5) has four distinct factors, so a(22) = 4.
%t A324923 difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
%t A324923 Table[Length[Union[difac[n]]],{n,100}]
%o A324923 (PARI)
%o A324923 A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
%o A324923 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
%Y A324923 One less than A317713.
%Y A324923 Cf. A000081, A003963, A061773, A061775, A109082, A109129, A120383, A196050.
%Y A324923 Cf. A324850, A324922, A324924, A324925, A324931, A324934, A324935, A324936, A366385, A366386.
%K A324923 nonn
%O A324923 1,3
%A A324923 _Gus Wiseman_, Mar 20 2019
%E A324923 Data section extended up to a(108) by _Antti Karttunen_, Oct 23 2023