A211159 -id:A211159 - cp's OEIS frontend

A278648 Consider the set S of integers 1 through n. a(n) is the number of unordered ways in which three distinct elements {a, b, c} of S satisfy ab = cn.

0, 0, 0, 0, 0, 0, 2, 0, 2, 1, 4, 0, 8, 0, 6, 8, 7, 0, 13, 0, 16, 12, 10, 0, 26, 6, 12, 13, 24, 0, 38, 0, 23, 20, 16, 24, 46, 0, 18, 24, 50, 0, 56, 0, 40, 49, 22, 0, 71, 15, 46, 32, 48, 0, 67, 40, 74, 36, 28, 0, 120, 0, 30, 73, 61, 48, 92, 0, 64, 44, 106, 0, 136, 0, 36, 86, 72, 60, 110

Offset: 0

Bobby Jacobs and Robert G. Wilson v, Nov 25 2016

Inspired by A278348.
Index of first occurrence of k >= 0, or zero if no such number exists: 1, 9, 6, 0, 10, 0, 14, 16, 12, 0, 22, 0, 21, 18, 0, 49, 20, 0, 38, 0, 33, 0, 46, 32, 28, 0, ..., ;
Numbers which never occur: 3, 5, 9, 11, 14, 17, 19, 21, 25, 27, 29, 31, 33, 34, 35, 37, 39, 41, 43, 47, 51, ..., ;
Records: 0, 2, 4, 8, 13, 16, 26, 38, 46, 50, 56, 71, 74, 120, 136, 176, 193, 214, 330, 355, 482, 574, 668, 839, 890, 996, 1088, 1223, 1528, 1920, 2039, 2224, 2374, 2646, 3055, 3120, 3811, 5010, 5539, 6208, 6591, 8566, 9139, 9690, 12359, 13894, 14796, 15331, 16118, 16558, 22048, ..., ;
which first occur for n: 0, 6, 10, 12, 18, 20, 24, 30, 36, 40, 42, 48, 56, 60, 72, 84, 90, 108, 120, 144, 168, 180, 210, 240, 280, 300, 330, 336, 360, 420, 480, 504, 540, 600, 630, 660, 720, 840, 1008, 1080, 1200, 1260, 1440, 1560, 1680, 1980, 2100, 2160, 2310, 2340, 2520, ..., .
If instead we look for the number of unordered ways two distinct elements {a, b} of S satisfy a*b = n, then a(n) = floor(sigma_0(n) - 2)) = A211159(n+1).
Number of 2 X 2 singular matrices of the form
  [c a]
  [b n]
with a, b, and c distinct positive integers less than n and a < b.

			a(6) = 2 since 2*3 = 1*6 and 3*4 = 2*6;
a(8) = 2 since 2*4 = 1*8 and 4*6 = 3*8;
a(9) = 1 since 3*6 = 2*9;
a(10) = 4 since 2*5 = 1*10, 4*5 = 2*10, 5*6 = 3*10, and 5*8 = 4*10;
a(12) = 8 since 2*6 = 1*12, 3*4 = 1*12, 3*8 = 2*12, 4*6 = 2*12, 4*9 = 3*12, 6*8 = 4*12, 6*10 = 5*12, and 8*9 = 6*12;
etc.

Bobby Jacobs and Robert G. Wilson v, Table of n, a(n) for n = 0..10000

Cf. A278348.

f[n_] := Block[{c = 0, k = 1}, While[k < n, c += Count[ Times @@@ Select[ Tuples[ Rest@ Most@ Divisors[k*n], 2], #[[1]] < #[[2]] < n &], k*n]; k++]; c]; Array[f, 52]

a(n) = (A278348(n) - A278348(n-1))/8.

a(p) = 0 for any prime p and for n: 0, 1 & 4.

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.

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

Original entry on oeis.org

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

A278648 Consider the set S of integers 1 through n. a(n) is the number of unordered ways in which three distinct elements {a, b, c} of S satisfy ab = cn.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

R

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

R

cp's OEIS Frontend

A278648 Consider the set S of integers 1 through n. a(n) is the number of unordered ways in which three distinct elements {a, b, c} of S satisfy a*b = c*n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

R

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

R

A278648 Consider the set S of integers 1 through n. a(n) is the number of unordered ways in which three distinct elements {a, b, c} of S satisfy ab = cn.