A122180 -id:A122180 - cp's OEIS frontend

A122181 Numbers k that can be written as k = xyz with 1 < x < y < z (A122180(k) > 0).

24, 30, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 128, 130, 132, 135, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 165, 168, 170, 174, 176, 180, 182, 184, 186, 189, 190, 192, 195

Offset: 1

Rick L. Shepherd, Aug 24 2006

Equivalently, numbers k with at least 7 divisors (A000005(k) > 6). Equivalently, numbers k with at least 5 proper divisors (A070824(k) > 4). Equivalently, numbers k such that i) k has at least three distinct prime factors (A000977), ii) k has two distinct prime factors and four or more total prime factors (k = p^j*q^m, p,q primes, j+m >= 4), or iii) k = p^m, a perfect power (A001597) but restricted to prime p and m >= 6 [= 1+2+3] (some terms of A076470).

			a(1) = 24 = 2*3*4, a product of three distinct proper divisors (omega(24) = 2, bigomega(24) = 4).
a(2) = 30 = 2*3*5, a product of three distinct prime factors (omega(30) = 3).
a(10) = 64 = 2*4*8 [= 2^1*2^2*2^3] (omega(64) = 1, bigomega(64) = 6).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A122179, A122180, A000005, A070824, A000977, A001221, A001222, A001597, A076470.

Select[Range[200], DivisorSigma[0, #] > 6 &] (* Amiram Eldar, Oct 05 2024 *)

PARI
```
isok(n) = numdiv(n)>6
```

isok(n) = (omega(n)==1 && bigomega(n)>5) || (omega(n)==2 && bigomega(n)>3) || (omega(n)>2)

A122179 Number of ways to write n as n = xyz with 1

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 3, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 0, 4, 0, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 4, 0, 0, 1, 3, 0, 1, 0, 1, 0, 1, 0, 6, 0, 0, 1, 1, 0, 1, 0, 4, 1, 0, 0, 4, 0, 0, 0, 2, 0, 4, 0, 1, 0, 0, 0, 6, 0, 1, 1, 3, 0, 1, 0, 2, 1

Offset: 1

Rick L. Shepherd, Aug 23 2006

nonn

x,y,z are proper factors of n. a(n) > 0 iff n is a term of A033942; a(n) = 0 iff n is a term of A037143.

			a(24) = 2 because 24 = 2*2*6 = 2*3*4, two products of three proper factors of 24.

Seiichi Manyama, Table of n, a(n) for n = 1..8192 (terms 1..2048 from Antti Karttunen)

Cf. A034836, A033942, A037143, A088432, A088433, A088434, A122180.

for(n=1,105, t=0; for(x=2,n-1, for(y=x,n-1, for(z=y,n-1, if(x*y*z==n, t++)))); print1(t,", "))

A122179(n) = { my(s=0); fordiv(n, x, if((x>1)&&(xAntti Karttunen, Aug 24 2017

A088434 Number of ways to write n as n = uvw with 1 <= u < v < w.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 0, 1, 1, 1, 0, 2, 0, 2, 1, 1, 0, 4, 0, 1, 1, 2, 0, 4, 0, 2, 1, 1, 1, 4, 0, 1, 1, 4, 0, 4, 0, 2, 2, 1, 0, 6, 0, 2, 1, 2, 0, 4, 1, 4, 1, 1, 0, 8, 0, 1, 2, 3, 1, 4, 0, 2, 1, 4, 0, 8, 0, 1, 2, 2, 1, 4, 0, 6, 1, 1, 0, 8, 1, 1, 1, 4, 0, 8, 1, 2, 1, 1, 1, 9, 0, 2, 2, 4, 0, 4, 0, 4, 4, 1, 0, 8, 0, 4, 1, 6, 0, 4, 1, 2, 2, 1, 1, 14

Offset: 1

Reinhard Zumkeller, Oct 01 2003

nonn

a(n)=0 iff n=1 or n prime or n prime^2: a(A000430(n)) = 0.

The integers a(n)+1 equal A045778(n) for n < 120 and differ at all n that admit factorization into 4 or more distinct factors, the smallest ones being n = 120 = 2*3*4*5, n = 144 = 2*3*4*6, n = 168 = 2*3*4*7, n = 180 = 2*3*5*6, ..., later continuing n = 312 = 2*3*4*13, n = 320 = 2*4*5*8, n = 324 = 2*3*6*9, n = 330 = 2*3*5*11, ... Coincidentally, A068350(5) to A068350(19) start this list. - R. J. Mathar, Jul 19 2007

			n=12: (1,2,6), (1,3,4): therefore a(12)=2;
n=18: (1,2,9), (1,3,6): therefore a(18)=2.

Antti Karttunen and Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 2048 terms from Antti Karttunen.)
Michael De Vlieger, Records and first positions of records in A088434.

Cf. A034836, A088432, A088433, A122179, A122180.

Table[Length[Select[Cases[Subsets[Divisors[n],{3}],{x_,y_,z_}->x*y*z],#==n &]],{n,102}] (* Jayanta Basu, May 23 2013 *)

A088434(n) = { my(s=0); fordiv(n, u, for(v=u+1, n-1, for(w=v+1, n, if(u*v*w==n, s++)))); (s); }; \\ Antti Karttunen, Aug 24 2017

Data section extended to 120 terms by Antti Karttunen, Aug 24 2017

A088432 Number of ways to write n as n = uvw with 1 <= u < v <= w.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 2, 0, 3, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 4, 0, 3, 1, 1, 1, 5, 0, 1, 1, 4, 0, 4, 0, 2, 2, 1, 0, 7, 1, 3, 1, 2, 0, 4, 1, 4, 1, 1, 0, 8, 0, 1, 2, 4, 1, 4, 0, 2, 1, 4, 0, 9, 0, 1, 3, 2, 1, 4, 0, 6, 2, 1, 0, 8, 1, 1, 1, 4, 0, 8, 1, 2, 1, 1, 1, 9, 0, 3, 2, 6, 0, 4, 0, 4, 4, 1, 0, 9, 0, 4, 1, 6, 0, 4, 1, 2, 2, 1, 1, 14

Offset: 1

Reinhard Zumkeller, Oct 01 2003

nonn

			n=12: (1,2,6), (1,3,4): therefore a(12)=2;
n=18: (1,2,9), (1,3,6), (2,3,3): therefore a(18)=3.
For n = p*q, p < q primes:  n = 1 * p * q, so a(n) = 1.
For n = p^2, p prime: n = 1 * p * p, so a(n) = 1.
For n = p^3, p prime: n = 1 * p * p^2, so a(n) = 1.
For n = p*q^2, p < q < p^2: n = 1 * p * pq = 1* q * p^2, so a(n) = 2 (see n=12).
For n = p*q^2, p < p^2 < q: n = 1 * p * pq = 1 * p^2 * q, so a(n) = 2
For n = p^4, p prime: n = 1 * p * p^3 = 1 * p^2 * p^2, so a(n) = 2.

Antti Karttunen, Table of n, a(n) for n = 1..3003

Cf. A034836, A088433, A088434, A122179, A122180, A323644.

a[n_] := Module[{s = 0}, Do[Do[Do[If[u v w == n, s++], {w, v, n}], {v, u + 1, n - 1}], {u, Divisors[n]}]; s];
Array[a, 120] (* Jean-François Alcover, Dec 10 2021, after Antti Karttunen *)

A088432(n) = { my(s=0); fordiv(n, u, for(v=u+1, n-1, for(w=v, n, if(u*v*w==n, s++)))); (s); }; \\ Antti Karttunen, Aug 24 2017

a(n) = 0 iff n=1 or n is prime: a(A008578(n)) = 0, a(A002808(n)) > 0.
a(n) = 1 iff n has 3 or 4 divisors (A323644) (see examples). - Bernard Schott, Dec 13 2021
a(n) = 2 if n = p^2*q, pA096156) or n = p^4 (A030514) (see examples). - Bernard Schott, Dec 16 2021

Data section extended to 120 terms by Antti Karttunen, Aug 24 2017

A088433 Number of ways to write n as n = uvw with 1<=u<=v

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 3, 1, 4, 2, 2, 1, 6, 1, 2, 2, 4, 1, 5, 1, 4, 2, 2, 2, 7, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 8, 1, 3, 2, 4, 1, 6, 2, 6, 2, 2, 1, 10, 1, 2, 4, 5, 2, 5, 1, 4, 2, 5, 1, 11, 1, 2, 3, 4, 2, 5, 1, 9, 3, 2, 1, 10, 2, 2, 2, 6, 1, 10, 2, 4, 2, 2, 2, 12, 1, 3, 4, 6

Offset: 1

Reinhard Zumkeller, Oct 01 2003

nonn

a(n) = 1 iff n prime or n prime^2: a(A000430(n))=1.

			n=12: (1,1,12), (1,2,6), (1,3,4), (2,2,3): therefore a(12)=4;
n=18: (1,1,18), (1,2,9), (1,3,6): therefore a(18)=3.

Antti Karttunen, Table of n, a(n) for n = 1..2048

Cf. A034836, A088432, A088434, A122179, A122180.

A088433(n) = { my(s=0); fordiv(n, u, for(v=u, n-1, for(w=v+1, n, if(u*v*w==n, s++)))); (s); }; \\ Antti Karttunen, Aug 24 2017

A200214 Ordered factorizations of n with 3 distinct parts, all > 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, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, 0, 0, 6, 0, 6, 0, 0, 0, 0, 0, 12, 0, 0, 0, 0, 0, 6, 0, 6, 0, 0, 0, 18, 0, 0, 0, 6, 0, 6, 0, 0, 0, 6, 0, 18, 0, 0, 0, 0, 0, 6, 0, 12, 0, 0, 0, 18

Offset: 1

Peter Luschny, Nov 14 2011

nonn

			a(24) = 6 = card({{2,3,4}, {2,4,3}, {3,2,4}, {3,4,2}, {4,2,3}, {4,3,2}}).
a(64) = 6 = card({{2,4,8}, {2,8,4}, {4,2,8}, {4,8,2}, {8,2,4}, {8,4,2}}).

Antti Karttunen, Table of n, a(n) for n = 1..1001
Benny Chor, Paul Lemke, Ziv Mador, On the number of ordered factorizations of natural numbers, Discrete Mathematics, Vol. 214[1], 2000, p. 123-133.
Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1), 2006.

Cf. A025487, A122180, A200213.

OrderedFactorizations[1] = {{}}; OrderedFactorizations[n_?PrimeQ] := {{n}}; OrderedFactorizations[n_] := OrderedFactorizations[n] = Flatten[Function[d, Prepend[#, d] & /@ OrderedFactorizations[n/d]] /@ Rest[Divisors[n]], 1]; a[n_] := With[{of3 = Sort /@ Select[OrderedFactorizations[n], Length[#] == 3 && Length[# // Union] == 3 &] // Union}, Length[Permutations /@ of3 // Flatten[#, 1] &]];  Table[a[n], {n, 1, 84}] (* Jean-François Alcover, Jul 02 2013, copied and adapted from The Mathematica Journal *)

A200214(n) = { my(s=0); fordiv(n, x, if((x>1)&&(xA122180, still quite naive) - Antti Karttunen, Jul 09 2017

a(n) = 6*A122180(n). - Antti Karttunen, Jul 08 2017

Description clarified, term a(0) removed and a second example added by Antti Karttunen, Jul 09 2017

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

A122181 Numbers k that can be written as k = x*y*z with 1 < x < y < z (A122180(k) > 0).

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A122179 Number of ways to write n as n = x*y*z with 1

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A088434 Number of ways to write n as n = u*v*w with 1 <= u < v < w.

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A088432 Number of ways to write n as n = u*v*w with 1 <= u < v <= w.

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A088433 Number of ways to write n as n = u*v*w with 1<=u<=v

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A200214 Ordered factorizations of n with 3 distinct parts, all > 1.

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Mathematica

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Extensions

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

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R

A122181 Numbers k that can be written as k = xyz with 1 < x < y < z (A122180(k) > 0).

A122179 Number of ways to write n as n = xyz with 1

A088434 Number of ways to write n as n = uvw with 1 <= u < v < w.

A088432 Number of ways to write n as n = uvw with 1 <= u < v <= w.

A088433 Number of ways to write n as n = uvw with 1<=u<=v