A122179 -id:A122179 - cp's OEIS frontend

A122180 Number of ways to write n as n = xyz with 1 < x < y < z < n.

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

Offset: 1

Rick L. Shepherd, Aug 23 2006

nonn

x,y,z are distinct proper factors of n. See A122181 for n such that a(n) > 0.

If n has at most five divisors then a(n) = 0. - David A. Corneth, Oct 24 2024

			a(48) = 2 because 48 = 2*3*8 = 2*4*6, two products of three distinct proper factors of 48.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1001 terms from Antti Karttunen)
Index entries for sequences computed from exponents in factorization of n

Cf. A034836, A088432, A088433, A088434, A122179, A122181, A200214.

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

A122180(n) = { my(s=0); fordiv(n, x, if((x>1)&&(xAntti Karttunen, Jul 08 2017

a(n) = {
	my(d = divisors(n));
	if(#d <= 5, return(0));
	my(res = 0, q);
	for(i = 2, #d,
		q = d[#d + 1 - i];
		if(d[i]^2 > q,
			return(res)
		);
		for(j = i + 1, #d,
			qj = q/d[j];
			if(qj <= d[j],
				next(2)
			);
			if(denominator(qj) == 1 && n % qj == 0,
				res++
			);
		);
	);
	res
} \\ David A. Corneth, Oct 24 2024

a(n) = A200214(n)/6. - Antti Karttunen, Jul 08 2017

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

Original entry on oeis.org

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)

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

A321376 Expansion of Product_{1 < i <= j <= k} (1 - x^(ijk)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, -1, 0, 1, -1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 1, 1, 4, 0, 0, 0, 2, 0, -1, 1, 2, 0, 0, -1, 2, 1, -1, -1, -1, 0, -2, 0, -4, -1, 2, -2, -3, 0, -4, -4, -1, -1, -6, -2, -7, 0, 2, -2, -10

Offset: 0

Seiichi Manyama, Nov 08 2018

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A122179, A321361, A321374, A321375, A321377.

G.f.: Product_{k>0} (1 - x^k)^A122179(k).

A347709 Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.

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, 2, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 0, 4, 0, 0, 1, 1, 0, 1, 0, 3, 1, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 4, 0, 1, 1, 2, 0, 1, 0, 2, 1, 0, 0, 4, 0, 1, 0, 3, 0, 1, 0, 1, 1, 0, 0, 5

Offset: 1

Gus Wiseman, Oct 14 2021

nonn

This is also the number of distinct possible alternating products of length-3 factorizations of n, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)), and where a factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			Representative factorizations for each of the a(360) = 9 alternating products:
   (2,2,90) -> 90
   (2,3,60) -> 40
   (2,4,45) -> 45/2
   (2,5,36) -> 72/5
   (2,6,30) -> 10
   (2,9,20) -> 40/9
  (2,10,18) -> 18/5
  (2,12,15) -> 5/2
   (3,8,15) -> 45/8

Antti Karttunen, Table of n, a(n) for n = 1..65537

Allowing factorizations of any length <= 3 gives A033273.
Positions of positive terms are A033942.
Positions of 0's are A037143.
The length-2 version is A072670.
Allowing any length gives A347460, reverse A038548.
Allowing any odd length gives A347708.
A001055 counts factorizations (strict A045778, ordered A074206).
A122179 counts length-3 factorizations.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions, positive A276024.
Cf. A000040, A001358, A002033, A046951, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456, A347461.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@Select[facs[n],Length[#]==3&]]],{n,100}]

A347709(n) = { my(rats=List([])); fordiv(n,z,my(yx=n/z); fordiv(yx, y, my(x = yx/y); if((y <= z) && (x <= y) && (x > 1), listput(rats,x*z/y)))); #Set(rats); }; \\ Antti Karttunen, Jan 29 2025

More terms from Antti Karttunen, Jan 29 2025

A321374 Expansion of Product_{1 < i <= j <= k} (1 + x^(ijk)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 2, 0, 0, 0, 3, 0, 1, 1, 3, 0, 2, 0, 5, 0, 1, 1, 8, 0, 3, 1, 8, 0, 5, 1, 12, 2, 4, 2, 18, 0, 9, 3, 18, 2, 14, 3, 28, 3, 13, 5, 39, 2, 22, 10, 42, 5, 32, 8, 61, 8, 34, 14, 85, 7, 52, 22, 91, 14, 74, 20, 132, 24, 80

Offset: 0

Seiichi Manyama, Nov 08 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A122179, A211857, A321359.

G.f.: Product_{k>0} (1 + x^k)^A122179(k).

A321375 Expansion of Product_{1 < i <= j <= k} 1/(1 - x^(ijk)).

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 2, 0, 0, 0, 5, 0, 1, 1, 4, 0, 2, 0, 9, 0, 2, 1, 12, 0, 3, 1, 16, 0, 7, 2, 20, 2, 6, 2, 36, 0, 13, 5, 37, 2, 21, 4, 60, 3, 23, 9, 80, 4, 35, 14, 106, 5, 58, 16, 137, 12, 66, 22, 210, 10, 100, 40, 238, 22, 147

Offset: 0

Seiichi Manyama, Nov 08 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A122179, A182270, A321360.

Euler transform of A122179.

G.f.: Product_{k>0} 1/(1 - x^k)^A122179(k).

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.

cp's OEIS Frontend

A122180 Number of ways to write n as n = x*y*z with 1 < x < y < z < n.

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

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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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A321376 Expansion of Product_{1 < i <= j <= k} (1 - x^(i*j*k)).

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A347709 Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.

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A321374 Expansion of Product_{1 < i <= j <= k} (1 + x^(i*j*k)).

A122180 Number of ways to write n as n = xyz with 1 < x < y < z < n.

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

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

A321376 Expansion of Product_{1 < i <= j <= k} (1 - x^(ijk)).

A321374 Expansion of Product_{1 < i <= j <= k} (1 + x^(ijk)).

A321375 Expansion of Product_{1 < i <= j <= k} 1/(1 - x^(ijk)).