A122179 -id:A122179 - 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

A321379 Number of ways to write n as n = abc*d with 1 < a <= b <= c <= d < n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 4

Offset: 1

Seiichi Manyama, Nov 08 2018

nonn

This sequence is different from A101638.

If p is prime, a(p^k) = A026810(k). - Robert Israel, Nov 08 2018

			16 = 2*2*2*2. So a(16) = 1.
24 = 2*2*2*3. So a(24) = 1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A026810, A122179, A218320.

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for a from 2 to floor(N^(1/4)) do
  for b from a to floor((N/a)^(1/3)) do
    for c from b to floor((N/a/b)^(1/2)) do
      for d from c to N/(a*b*c) do
        V[a*b*c*d]:= V[a*b*c*d]+1
od od od od:
convert(V,list); # Robert Israel, Nov 08 2018

cp's OEIS Frontend

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

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A321379 Number of ways to write n as n = abc*d with 1 < a <= b <= c <= d < n.

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Maple

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

R

A321379 Number of ways to write n as n = a*b*c*d with 1 < a <= b <= c <= d < n.

Views

Author

Keywords

Comments

Examples

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Programs

Maple

A321379 Number of ways to write n as n = abc*d with 1 < a <= b <= c <= d < n.