A002033 -id:A002033 - cp's OEIS frontend

A251683 Irregular triangular array: T(n,k) is the number of ordered factorizations of n with exactly k factors, n >= 1, 1 <= k <= A086436(n).

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 2, 1, 2, 1, 3, 3, 1, 1, 1, 4, 3, 1, 1, 4, 3, 1, 2, 1, 2, 1, 1, 6, 9, 4, 1, 1, 1, 2, 1, 2, 1, 1, 4, 3, 1, 1, 6, 6, 1, 1, 4, 6, 4, 1, 1, 2, 1, 2, 1, 2, 1, 7, 12, 6, 1, 1, 2, 1, 2, 1, 6, 9, 4

Offset: 1

Geoffrey Critzer, Dec 06 2014

Row sums = A074206.
Row lengths give A086436.
T(n,2) = A070824(n).
T(n,3) = A200221(n).
Sum_{k>=1} k*T(n,k) = A254577.
For all n > 1,  Sum_{k=1..A086436(n)} (-1)^k*T(n,k) = A008683(n). - Geoffrey Critzer, May 25 2018
From Gus Wiseman, Aug 21 2020: (Start)
Also the number of strict length k + 1 chains of divisors from n to 1. For example, row n = 24 counts the following chains:
  24/1  24/2/1   24/4/2/1   24/8/4/2/1
        24/3/1   24/6/2/1   24/12/4/2/1
        24/4/1   24/6/3/1   24/12/6/2/1
        24/6/1   24/8/2/1   24/12/6/3/1
        24/8/1   24/8/4/1
        24/12/1  24/12/2/1
                 24/12/3/1
                 24/12/4/1
                 24/12/6/1
(End)

			Triangle T(n,k) begins:
  1;
  1;
  1;
  1, 1;
  1;
  1, 2;
  1;
  1, 2, 1;
  1, 1;
  1, 2;
  1;
  1, 4, 3;
  1;
  1, 2;
  1, 2;
  ...
There are 8 ordered factorizations of the integer 12: 12, 6*2, 4*3, 3*4, 2*6, 3*2*2, 2*3*2, 2*2*3.  So T(12,1)=1, T(12,2)=4, and T(12,3)=3.

Alois P. Heinz, Rows n = 1..4000, flattened
Jeffery Kline, On the eigenstructure of sparse matrices related to the prime number theorem, Linear Algebra and its Applications (2020) Vol. 584, 409-430.
Arnold Knopfmacher and Michael Mays, Ordered and Unordered Factorizations of Integers, The Mathematica Journal, Vol 10 (1).
Eric Weisstein's World of Mathematics, Ordered Factorization

Cf. A008683, A070824, A200221, A254577.
A008480 gives rows ends.
A086436 gives row lengths.
A124433 is the same except for signs and zeros.
A334996 is the same except for zeros.
A337107 is the restriction to factorial numbers (but with zeros).
A000005 counts divisors.
A001055 counts factorizations.
A001222 counts prime factors with multiplicity.
A074206 counts strict chains of divisors from n to 1.
A067824 counts strict chains of divisors starting with n.
A122651 counts strict chains of divisors summing to n.
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict nonempty chains of divisors of n.
A337071 counts strict chains of divisors starting with n!.
A337256 counts strict chains of divisors of n.
Cf. A001221, A002033, A124010, A167865, A337070, A337105.

with(numtheory):
b:= proc(n) option remember; expand(x*(1+
      add(b(n/d), d=divisors(n) minus {1, n})))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
seq(T(n), n=1..100);  # Alois P. Heinz, Dec 07 2014

f[1] = {{}};
f[n_] := f[n] =
  Level[Table[
    Map[Prepend[#, d] &, f[n/d]], {d, Rest[Divisors[n]]}], {2}];
Prepend[Map[Select[#, # > 0 &] &,
  Drop[Transpose[
    Table[Map[Count[#, k] &,
      Map[Length, Table[f[n], {n, 1, 40}], {2}]], {k, 1, 10}]],
   1]],{1}] // Grid
(* Second program: *)
b[n_] := b[n] = x(1+Sum[b[n/d], {d, Divisors[n]~Complement~{1, n}}]);
T[n_] := CoefficientList[b[n]/x, x];
Array[T, 100] // Flatten (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

Dirichlet g.f.: 1/(1 - y*(zeta(x)-1)).

A340101 Number of factorizations of 2n + 1 into odd factors > 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 2, 3, 1, 1, 2, 2, 1, 2, 1, 1, 4, 1, 2, 2, 1, 2, 2, 1, 1, 4, 2, 1, 2, 1, 1, 4, 2, 1, 5, 1, 2, 2, 1, 2, 2, 2, 1, 4, 1, 1, 5, 1, 1, 2, 1, 2, 4, 2, 2, 2, 3, 1, 2, 1, 2, 7, 1, 1, 2, 2, 2, 4, 1, 1, 4, 2, 1, 2, 2, 1, 5, 1, 2, 4, 1, 4, 2, 1, 1, 2, 2, 2, 7, 1, 1, 5, 1, 1, 2, 2, 2, 4, 2

Offset: 0

Gus Wiseman, Dec 28 2020

nonn

			The factorizations for 2n + 1 = 27, 45, 135, 225, 315, 405, 1155:
  27      45      135       225       315       405         1155
  3*9     5*9     3*45      3*75      5*63      5*81        15*77
  3*3*3   3*15    5*27      5*45      7*45      9*45        21*55
          3*3*5   9*15      9*25      9*35      15*27       33*35
                  3*5*9     15*15     15*21     3*135       3*385
                  3*3*15    5*5*9     3*105     5*9*9       5*231
                  3*3*3*5   3*3*25    5*7*9     3*3*45      7*165
                            3*5*15    3*3*35    3*5*27      11*105
                            3*3*5*5   3*5*21    3*9*15      3*5*77
                                      3*7*15    3*3*5*9     3*7*55
                                      3*3*5*7   3*3*3*15    5*7*33
                                                3*3*3*3*5   3*11*35
                                                            5*11*21
                                                            7*11*15
                                                            3*5*7*11

Antti Karttunen, Table of n, a(n) for n = 0..32768

The version for partitions is A160786, ranked by A300272.
The even version is A340785.
The odd-length case is A340102.
A000009 counts partitions into odd parts, ranked by A066208.
A001055 counts factorizations, with strict case A045778.
A027193 counts partitions of odd length, ranked by A026424.
A058695 counts partitions of odd numbers, ranked by A300063.
A316439 counts factorizations by product and length.
Cf. A000700, A002033, A027187, A028260, A074206, A078408, A174726, A236914, A320732, A339846.
Odd bisection of A001055, and also of A349907.

g:= proc(n, k) option remember; `if`(n>k, 0, 1)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> g(2*n+1$2):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 30 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OddQ[Times@@#]&]],{n,1,100,2}]

A001055(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m), s += A001055(n/d, d))); (s)); \\ After code in A001055
A340101(n) = A001055(n+n+1); \\ Antti Karttunen, Dec 13 2021

a(n) = A001055(2n+1).

a(n) = A349907(2n+1). - Antti Karttunen, Dec 13 2021

Data section extended up to 105 terms by Antti Karttunen, Dec 13 2021

A325780 Heinz numbers of perfect integer partitions.

Original entry on oeis.org

1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 234, 256, 260, 294, 392, 416, 486, 500, 512, 798, 1024, 1026, 1064, 1088, 1458, 1936, 2048, 2058, 2300, 2432, 2500, 2744, 3042, 3380, 4096, 4374, 4698, 5104, 5408, 5888, 8192, 8658, 9620, 10878

Offset: 1

Gus Wiseman, May 21 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

The sum of prime indices of n is A056239(n). A number is in this sequence iff all of its divisors have distinct sums of prime indices, and these sums cover an initial interval of nonnegative integers. For example, the divisors of 260 are {1, 2, 4, 5, 10, 13, 20, 26, 52, 65, 130, 260}, with respective sums of prime indices {0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 10, 11}, so 260 is in the sequence.

			The sequence of terms together with their prime indices begins:
      1: {}
      2: {1}
      4: {1,1}
      6: {1,2}
      8: {1,1,1}
     16: {1,1,1,1}
     18: {1,2,2}
     20: {1,1,3}
     32: {1,1,1,1,1}
     42: {1,2,4}
     54: {1,2,2,2}
     56: {1,1,1,4}
     64: {1,1,1,1,1,1}
    100: {1,1,3,3}
    128: {1,1,1,1,1,1,1}
    162: {1,2,2,2,2}
    176: {1,1,1,1,5}
    234: {1,2,2,6}
    256: {1,1,1,1,1,1,1,1}
    260: {1,1,3,6}

Equals the sorted concatenation of the triangle A258119.
A subsequence of A299702 and A325781.
Cf. A002033, A056239, A103300, A108917, A112798, A126796, A188431, A325763, A325768, A325782.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
Select[Range[1000],Sort[hwt/@Rest[Divisors[#]]]==Range[DivisorSigma[0,#]-1]&]

Intersection of A299702 (knapsack partitions) and A325781 (complete partitions).

A330973 Least positive integer with exactly n factorizations into factors > 1, and 0 if no such number exists.

Original entry on oeis.org

1, 4, 8, 12, 16, 0, 24, 0, 36, 0, 60, 48, 0, 0, 128, 72, 0, 0, 96, 0, 120, 256, 0, 0, 0, 180, 0, 0, 144, 192, 216, 0, 0, 0, 0, 420, 0, 240, 0, 0, 0, 1024, 0, 0, 384, 0, 288, 0, 0, 0, 0, 360, 0, 0, 0, 2048, 432, 0, 0, 0, 0, 0, 0, 480, 0, 900, 768, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 06 2020

nonn

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

All nonzero terms belong to A025487.
Includes all highly factorable numbers A033833.
Factorizations are A001055, with image A045782.
The version without zeros is A045783.
The sorted version is A330972.
The strict version is A330974.
Positions of zeros are A330976.
Cf. A001055, A001222, A002033, A007716, A045778, A045779, A330935, A330992, A330997, A330998, A346426.

nn=10;
fam[n_]:=fam[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[fam[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nds=Length/@Array[fam[#]&,2^nn];
Table[If[#=={},0,#[[1,1]]]&[Position[nds,i]],{i,nn}]

More terms from Jinyuan Wang, Jul 07 2021

A174725 a(n) = (A074206(n) + A008683(n))/2.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 2, 1, 2, 0, 4, 0, 2, 2, 4, 0, 4, 0, 4, 2, 2, 0, 10, 1, 2, 2, 4, 0, 6, 0, 8, 2, 2, 2, 13, 0, 2, 2, 10, 0, 6, 0, 4, 4, 2, 0, 24, 1, 4, 2, 4, 0, 10, 2, 10, 2, 2, 0, 22, 0, 2, 4, 16, 2, 6, 0, 4, 2, 6, 0, 38, 0, 2, 4, 4, 2

Offset: 1

Mats Granvik, Mar 28 2010

nonn

From Mats Granvik, May 25 2017: (Start)
A074206(n) = A002033(n-1) = a(n) + A174726(n).
A008683(n) = a(n) - A174726(n).
Let m = size of matrix a matrix T, and let T be defined as follows:
T(n,k) = if m = 1 then 1 else if mod(n, k) = 0 then if and(n = k, n = m) then 0 else 1 else if and(n = 1, k = m) then 1 else 0
a(n) is then the number of permutation matrices with a positive contribution in the determinant of matrix T. The determinant of T is equal to the Möbius function A008683, see Mathematica program below for how to compute the determinant.
A174726 is the number of permutation matrices with a negative contribution in the determinant of matrix T.
(End)
From Gus Wiseman, Jan 04 2021: (Start)
Also the number of ordered factorizations of n into an even number of factors > 1. The non-ordered case is A339846. For example, the a(n) factorizations for n = 12, 24, 30, 32, 36 are:
  (2*6)  (3*8)      (5*6)   (4*8)      (4*9)
  (3*4)  (4*6)      (6*5)   (8*4)      (6*6)
  (4*3)  (6*4)      (10*3)  (16*2)     (9*4)
  (6*2)  (8*3)      (15*2)  (2*16)     (12*3)
         (12*2)     (2*15)  (2*2*2*4)  (18*2)
         (2*12)     (3*10)  (2*2*4*2)  (2*18)
         (2*2*2*3)          (2*4*2*2)  (3*12)
         (2*2*3*2)          (4*2*2*2)  (2*2*3*3)
         (2*3*2*2)                     (2*3*2*3)
         (3*2*2*2)                     (2*3*3*2)
                                       (3*2*2*3)
                                       (3*2*3*2)
                                       (3*3*2*2)
(End)

Mats Granvik, Table of n, a(n) for n = 1..10000

Cf. A008683, A051731.
The odd version is A174726.
The unordered version is A339846.
A001055 counts factorizations, with strict case A045778.
A058696 counts partitions of even numbers, ranked by A300061.
A074206 counts ordered factorizations, with strict case A254578.
A251683 counts ordered factorizations by product and length.
Other cases of even length:
- A024430 counts set partitions of even length.
- A027187 counts partitions of even length.
- A034008 counts compositions of even length.
- A052841 counts ordered set partitions of even length.
- A067661 counts strict partitions of even length.
- A332305 counts strict compositions of even length
Cf. A002033, A027193, A028260, A050320, A058695, A236913, A316439, A320655, A320656, A339890, A378216 (Dirichlet inverse).

(* From Mats Granvik, May 25 2017: (Start) *)
Clear[t, nn]; nn = 77; t[1, 1] = 1; t[n_, k_] := t[n, k] = If[k == 1, Sum[t[n, k + i], {i, 1, n - 1}], If[Mod[n, k] == 0, t[n/k, 1], 0], 0]; Monitor[Table[Sum[If[Mod[n, k] == 0, MoebiusMu[k]*t[n/k, 1], 0], {k, 1, 77}], {n, 1, nn}], n]
(* The Möbius function as a determinant *) Table[Det[Table[Table[If[m == 1, 1, If[Mod[n, k] == 0, If[And[n == k, n == m], 0, 1], If[And[n == 1, k == m], 1, 0]]], {k, 1, m}], {n, 1, m}]], {m, 1, 42}]
(* (End) *)
ordfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@ordfacs[n/d],{d,Rest[Divisors[n]]}]];
Table[Length[Select[ordfacs[n],EvenQ@*Length]],{n,100}] (* Gus Wiseman, Jan 04 2021 *)

a(n) = (Mobius transform of a(n)) + (Mobius transform of A174726). - Mats Granvik, Apr 04 2010
From Mats Granvik, May 25 2017: (Start)
This sequence is the Moebius transform of A074206.
a(n) = (A074206(n) + A008683(n))/2.
(End)
G.f. A(x) satisfies: A(x) = x + Sum_{i>=2} Sum_{j>=2} A(x^(i*j)). - Ilya Gutkovskiy, May 11 2019

References to A002033(n-1) changed to A074206(n) by Antti Karttunen, Nov 23 2024

A334996 Irregular triangle read by rows: T(n, m) is the number of ways to distribute Omega(n) objects into precisely m distinct boxes, with no box empty (Omega(n) >= m).

Original entry on oeis.org

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

Offset: 1

Stefano Spezia, May 19 2020

n is the specification number for a set of Omega(n) objects (see Theorem 3 in Beekman's article).
The specification number of a multiset is also called its Heinz number. - Gus Wiseman, Aug 25 2020
From Gus Wiseman, Aug 25 2020: (Start)
For n > 1, T(n,k) is also the number of ordered factorizations of n into k factors > 1. For example, row n = 24 counts the following ordered factorizations (the first column is empty):
  24  3*8   2*2*6  2*2*2*3
      4*6   2*3*4  2*2*3*2
      6*4   2*4*3  2*3*2*2
      8*3   2*6*2  3*2*2*2
      12*2  3*2*4
      2*12  3*4*2
            4*2*3
            4*3*2
            6*2*2
For n > 1, T(n,k) is also the number of strict length-k chains of divisors from n to 1. For example, row n = 36 counts the following chains (the first column is empty):
  36/1  36/2/1   36/4/2/1   36/12/4/2/1
        36/3/1   36/6/2/1   36/12/6/2/1
        36/4/1   36/6/3/1   36/12/6/3/1
        36/6/1   36/9/3/1   36/18/6/2/1
        36/9/1   36/12/2/1  36/18/6/3/1
        36/12/1  36/12/3/1  36/18/9/3/1
        36/18/1  36/12/4/1
                 36/12/6/1
                 36/18/2/1
                 36/18/3/1
                 36/18/6/1
                 36/18/9/1
(End)

			The triangle T(n, m) begins
  n\m| 0     1     2     3     4
  ---+--------------------------
   1 | 0
   2 | 0     1
   3 | 0     1
   4 | 0     1     1
   5 | 0     1
   6 | 0     1     2
   7 | 0     1
   8 | 0     1     2     1
   9 | 0     1     1
  10 | 0     1     2
  11 | 0     1
  12 | 0     1     4     3
  13 | 0     1
  14 | 0     1     2
  15 | 0     1     2
  16 | 0     1     3     3     1
  ...
From _Gus Wiseman_, Aug 25 2020: (Start)
Row n = 36 counts the following distributions of {1,1,2,2} (the first column is empty):
  {1122}  {1}{122}  {1}{1}{22}  {1}{1}{2}{2}
          {11}{22}  {1}{12}{2}  {1}{2}{1}{2}
          {112}{2}  {11}{2}{2}  {1}{2}{2}{1}
          {12}{12}  {1}{2}{12}  {2}{1}{1}{2}
          {122}{1}  {12}{1}{2}  {2}{1}{2}{1}
          {2}{112}  {1}{22}{1}  {2}{2}{1}{1}
          {22}{11}  {12}{2}{1}
                    {2}{1}{12}
                    {2}{11}{2}
                    {2}{12}{1}
                    {2}{2}{11}
                    {22}{1}{1}
(End)

Richard Beekman, An Introduction to Number-Theoretic Combinatorics, Lulu Press 2017.

Stefano Spezia, First 3000 rows of the table, flattened
Richard Beekman, A General Solution of the Ferris Wheel Problem.

Cf. A000007 (1st column), A000012 (2nd column), A001222 (Omega function), A002033 (row sums shifted left), A007318.
A008480 gives rows ends.
A073093 gives row lengths.
A074206 gives row sums.
A112798 constructs the multiset with each specification number.
A124433 is a signed version.
A251683 is the version with zeros removed.
A334997 is the non-strict version.
A337107 is the restriction to factorial numbers.
A001055 counts factorizations.
A067824 counts strict chains of divisors starting with n.
A122651 counts strict chains of divisors summing to n.
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict chains of divisors.
A337105 counts strict chains of divisors from n! to 1.
Cf. A007425, A008683, A056239, A124010, A167865, A317144, A319001.

tau[n_,k_]:=If[n==1,1,Product[Binomial[Extract[Extract[FactorInteger[n],i],2]+k,k],{i,1,Length[FactorInteger[n]]}]]; (* A334997 *)
T[n_,m_]:=Sum[(-1)^k*Binomial[m,k]*tau[n,m-k-1],{k,0,m-1}]; Table[T[n,m],{n,1,30},{m,0,PrimeOmega[n]}]//Flatten
(* second program *)
chc[n_]:=If[n==1,{{}},Prepend[Join@@Table[Prepend[#,n]&/@chc[d],{d,DeleteCases[Divisors[n],1|n]}],{n}]]; (* change {{}} to {} if a(1) = 0 *)
Table[Length[Select[chc[n],Length[#]==k&]],{n,30},{k,0,PrimeOmega[n]}] (* Gus Wiseman, Aug 25 2020 *)

TT(n, k) = if (k==0, 1, sumdiv(n, d, TT(d, k-1))); \\ A334997
T(n, m) = sum(k=0, m-1, (-1)^k*binomial(m, k)*TT(n, m-k-1));
tabf(nn) = {for (n=1, nn, print(vector(bigomega(n)+1, k, T(n, k-1))););} \\ Michel Marcus, May 20 2020

T(n, m) = Sum_{k=0..m-1} (-1)^k*binomial(m,k)*tau_{m-k-1}(n), where tau_s(r) = A334997(r, s) (see Theorem 3, Lemma 1 and Lemma 2 in Beekman's article).
Conjecture: Sum_{m=0..Omega(n)} T(n, m) = A002033(n-1) for n > 1.
The above conjecture is true since T(n, m) is also the number of ordered factorizations of n into m factors (see Comments) and A002033(n-1) is the number of ordered factorizations of n. - Stefano Spezia, Aug 21 2025

A336500 Number of divisors d|n with distinct prime multiplicities such that the quotient n/d also has distinct prime multiplicities.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(1) = 1 through a(16) = 5 divisors:
  1  1  1  1  1  2  1  1  1  2  1  1  1  2  3  1
     2  3  2  5  3  7  2  3  5 11  3 13  7  5  2
           4           4  9        4           4
                       8          12           8
                                              16

A336419 is the version for superprimorials.
A336568 gives positions of zeros.
A336869 is the restriction to factorials.
A007425 counts divisors of divisors.
A056924 counts divisors greater than their quotient.
A074206 counts chains of divisors from n to 1.
A130091 lists numbers with distinct prime exponents.
A181796 counts divisors with distinct prime multiplicities.
A336424 counts factorizations using A130091.
A336422 counts divisible pairs of divisors, both in A130091.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336423 counts chains in A130091, with maximal version A336569.
A336568 gives numbers not a product of two elements of A130091.
A336571 counts divisor sets using A130091, with maximal version A336570.
Cf. A000005, A001055, A002033, A098859, A124010, A167865, A253249, A336870.

Table[Length[Select[Divisors[n],UnsameQ@@Last/@FactorInteger[#]&&UnsameQ@@Last/@FactorInteger[n/#]&]],{n,25}]

A340102 Number of factorizations of 2n + 1 into an odd number of odd factors > 1.

Original entry on oeis.org

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

Offset: 0

Gus Wiseman, Dec 30 2020

nonn

			The factorizations for 2n + 1 = 135, 225, 315, 405, 675, 1155, 1215:
  135      225      315      405         675         1155      1215
  3*5*9    5*5*9    5*7*9    5*9*9       3*3*75      3*5*77    3*5*81
  3*3*15   3*3*25   3*3*35   3*3*45      3*5*45      3*7*55    3*9*45
           3*5*15   3*5*21   3*5*27      3*9*25      5*7*33    5*9*27
                    3*7*15   3*9*15      5*5*27      3*11*35   9*9*15
                             3*3*3*3*5   5*9*15      5*11*21   3*15*27
                                         3*15*15     7*11*15   3*3*135
                                         3*3*3*5*5             3*3*3*5*9
                                                               3*3*3*3*15

The version for partitions is A160786, ranked by A300272.
The not necessarily odd-length version is A340101.
A000009 counts partitions into odd parts, ranked by A066208.
A001055 counts factorizations, with strict case A045778.
A027193 counts partitions of odd length, ranked by A026424.
A058695 counts partitions of odd numbers, ranked by A300063.
A316439 counts factorizations by product and length.
Cf. A000700, A002033, A027187, A028260, A074206, A078408, A174726, A236914, A320732, A339846.

g:= proc(n, k, t) option remember; `if`(n>k, 0, t)+
      `if`(isprime(n), 0, add(`if`(d>k, 0, g(n/d, d, 1-t)),
          d=numtheory[divisors](n) minus {1, n}))
    end:
a:= n-> `if`(n=0, 0, g(2*n+1$2, 1)):
seq(a(n), n=0..100);  # Alois P. Heinz, Dec 30 2020

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OddQ[Length[#]]&&OddQ[Times@@#]&]],{n,1,100,2}];

A342083 Number of chains of strictly inferior divisors from n to 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 1, 3, 2, 3, 1, 5, 2, 4, 2, 2, 1, 7, 1, 2, 3, 3, 2, 5, 1, 3, 2, 4, 1, 8, 1, 2, 3, 3, 2, 5, 1, 6, 2, 2, 1, 7, 2, 2, 2

Offset: 1

Gus Wiseman, Feb 28 2021

nonn

We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.
These chains have first-quotients (in analogy with first-differences) that are term-wise > their decapitation (maximum element removed). Equivalently, x > y^2 for all adjacent x, y. For example, the divisor chain q = 60/6/2/1 has first-quotients (10,3,2), which are > (6,2,1), so q is counted under a(60).
Also the number of factorizations of n where each factor is greater than the product of all previous factors.

			The a(n) chains for n = 2, 6, 12, 24, 42, 48, 60, 72:
  2/1  6/1    12/1    24/1    42/1      48/1      60/1      72/1
       6/2/1  12/2/1  24/2/1  42/2/1    48/2/1    60/2/1    72/2/1
              12/3/1  24/3/1  42/3/1    48/3/1    60/3/1    72/3/1
                      24/4/1  42/6/1    48/4/1    60/4/1    72/4/1
                              42/6/2/1  48/6/1    60/5/1    72/6/1
                                        48/6/2/1  60/6/1    72/8/1
                                                  60/6/2/1  72/6/2/1
                                                            72/8/2/1
The a(n) factorizations for n = 2, 6, 12, 24, 42, 48, 60, 72:
  2  6    12   24    42     48     60      72
     2*3  2*6  3*8   6*7    6*8    2*30    8*9
          3*4  4*6   2*21   2*24   3*20    2*36
               2*12  3*14   3*16   4*15    3*24
                     2*3*7  4*12   5*12    4*18
                            2*3*8  6*10    6*12
                                   2*3*10  2*4*9
                                           2*3*12

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

The restriction to powers of 2 is A040039.
Not requiring strict inferiority gives A074206 (ordered factorizations).
The weakly inferior version is A337135.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A342098, or A000929 allowing equality.
A000005 counts divisors.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
A342086 counts chains of divisors with strictly increasing quotients > 1.
- Inferior: A033676, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341673.
Cf. A000203, A001248, A002033, A006530, A018819, A020639, A045690, A337105, A342087, A342094, A342095, A342096, A342097.

cen[n_]:=If[n==1,{{1}},Prepend[#,n]&/@Join@@cen/@Select[Divisors[n],#

G.f.: x + Sum_{k>=1} a(k) * x^(k*(k + 1)) / (1 - x^k). - Ilya Gutkovskiy, Nov 03 2021

A336568 Numbers that are not a product of two numbers each having distinct prime multiplicities.

Original entry on oeis.org

30, 42, 66, 70, 78, 102, 105, 110, 114, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 210, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 330, 345, 354, 357, 366, 370, 374, 385, 390, 399, 402, 406, 410, 418, 420, 426, 429

Offset: 1

Gus Wiseman, Aug 06 2020

nonn

First differs from A007304 and A093599 in having 210.
First differs from A287483 in having 222.
First differs from A350352 in having 420.
A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			Selected terms together with their prime indices:
   660: {1,1,2,3,5}
   798: {1,2,4,8}
   840: {1,1,1,2,3,4}
  3120: {1,1,1,1,2,3,6}
  9900: {1,1,2,2,3,3,5}

A336500 has zeros at these positions.
A007425 counts divisors of divisors.
A056924 counts divisors greater than their quotient.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A336424 counts factorizations using A130091.
A336422 counts divisible pairs of divisors, both in A130091.
A327498 is the maximum divisor with distinct prime multiplicities.
A336423 counts chains in A130091, with maximal version A336569.
A336571 counts divisor sets using A130091, with maximal version A336570.
Cf. A001055, A002033, A098859, A124010, A167865, A253249, A336420.

strsig[n_]:=UnsameQ@@Last/@FactorInteger[n]
Select[Range[100],Function[n,Select[Divisors[n],strsig[#]&&strsig[n/#]&]=={}]]

cp's OEIS Frontend

A251683 Irregular triangular array: T(n,k) is the number of ordered factorizations of n with exactly k factors, n >= 1, 1 <= k <= A086436(n).

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A340101 Number of factorizations of 2n + 1 into odd factors > 1.

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A325780 Heinz numbers of perfect integer partitions.

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A330973 Least positive integer with exactly n factorizations into factors > 1, and 0 if no such number exists.

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A174725 a(n) = (A074206(n) + A008683(n))/2.

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A334996 Irregular triangle read by rows: T(n, m) is the number of ways to distribute Omega(n) objects into precisely m distinct boxes, with no box empty (Omega(n) >= m).

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A336500 Number of divisors d|n with distinct prime multiplicities such that the quotient n/d also has distinct prime multiplicities.

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A340102 Number of factorizations of 2n + 1 into an odd number of odd factors > 1.

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