Jacob Sprittulla - cp's OEIS frontend

User: Jacob Sprittulla

Jacob Sprittulla has authored 2 sequences.

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

A334739 Number of unordered factorizations of n with 2 different parts > 1.

Original entry on oeis.org

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, 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, 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

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

Jacob Sprittulla, Table of n, a(n) for n = 1..1000
Index to sequences related to prime signature

Cf. A334740 (3 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

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

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.

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User: Jacob Sprittulla

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

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