A334739 -id:A334739 - cp's OEIS frontend

A334740 Number of unordered factorizations of n with 3 different parts > 1.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 4, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 1, 0, 3, 0, 0, 0, 4, 0, 0, 0, 1, 0, 4, 0, 0, 0, 0, 0, 8, 0, 0, 0, 1

Offset: 1

Jacob Sprittulla, May 09 2020

nonn

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).

			a(48) = 3 = #{ (6,4,2), (8,3,2), (4,3,2,2) }.

Jacob Sprittulla, Table of n, a(n) for n = 1..1000

Cf. A334739 (2 different parts), A072670 (2 parts), A122179 (3 parts), A211159 (2 distinct parts), A122180 (3 distinct parts), A001055, A045778

maxe  <- function(n, d)  { i=0; while( n%%(d^(i+1))==0 )  { i=i+1 }; i }
uhRec <- function(n, l=1)  {
  uh = 0
  if( n<=0 ) {
    return(0)
  } else if(n==1) {
    return(ifelse(l==0, 1, 0))
  } else if(l<=0) {
    return(0)
  } else if( (n>=2) && (l>=1) )  {
    for(d in 2:n)  {
      m = maxe(n, d)
      if(m>=1)  for(i in 1:m)  for(j in 1:min(i, l))   {
        uhj = uhRec( n/d^i, l-j )
        uh  = uh +  log(d)/log(n) * (-1)^(j+1) * choose(i, j) * uhj
      }
    }
    return(round(uh, 3))
  }
}
n=100; l=2; sapply(1:n, uhRec, l)    # A334739
n=100; l=3; sapply(1:n, uhRec, l)    # A334740

cp's OEIS Frontend

A334740 Number of unordered factorizations of n with 3 different parts > 1.

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