A334739 - OEIS

%I #25 Jan 03 2021 21:15:52

%S 0,0,0,0,0,1,0,1,0,1,0,3,0,1,1,2,0,3,0,3,1,1,0,5,0,1,1,3,0,3,0,5,1,1,

%T 1,6,0,1,1,5,0,3,0,3,3,1,0,8,0,3,1,3,0,5,1,5,1,1,0,6,0,1,3,6,1,3,0,3,

%U 1,3,0,10,0,1,3,3,1,3,0,8,2,1,0,6,1,1,1,5,0,6,1,3,1,1,1,10,0,3,3,6

%N Number of unordered factorizations of n with 2 different parts > 1.

%C a(n) depends only on the prime signature of n. E.g., a(12)=a(75), since 12=2^2*3 and 75=5^2*3 share the same prime signature (2,1).

%H Jacob Sprittulla, <a href="/A334739/b334739.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>

%F (Joint) D.g.f.: Product_{n>=2} ( 1 + t/(n^s-1) ).

%F Recursion: a(n) = h_2(n), where h_l(n) * log(n) = Sum_{ d^i | n } Sum_{j=1..l} (-1)^(j+1) * h_{l-j}(n/d^i) * log(d), with h_l(n)=1 if n=1 and l=0 otherwise h_l(n)=0.

%e a(24) = 5 = #{ (12,2), (6,4), (8,3), (6,2,2), (3,2,2,2) }.

%o (R)

%o maxe  <- function(n,d)  { i=0; while( n%%(d^(i+1))==0 )  { i=i+1 }; i }

%o uhRec <- function(n,l=1)  {

%o   uh = 0

%o   if( n<=0 ) {

%o     return(0)

%o   } else if(n==1) {

%o     return(ifelse(l==0,1,0))

%o   } else if(l<=0) {

%o     return(0)

%o   } else if( (n>=2) && (l>=1) )  {

%o     for(d in 2:n)  {

%o       m = maxe(n,d)

%o       if(m>=1)  for(i in 1:m)  for(j in 1:min(i,l))   {

%o         uhj = uhRec( n/d^i, l-j )

%o         uh  = uh +  log(d)/log(n) * (-1)^(j+1) * choose(i,j) * uhj

%o       }

%o     }

%o     return(round(uh,3))

%o   }

%o }

%o n=100; l=2; sapply(1:n,uhRec,l)    # A334739

%o n=100; l=3; sapply(1:n,uhRec,l)    # A334740

%Y Cf. A334740 (3 different parts), A072670 (2 parts), A122179 (3 parts), A211159 (2 distinct parts), A122180 (3 distinct parts), A001055, A045778.

%K nonn

%O 1,12

%A _Jacob Sprittulla_, May 09 2020