A316439 -id:A316439 - cp's OEIS frontend

A001055 The multiplicative partition function: number of ways of factoring n with all factors greater than 1 (a(1) = 1 by convention).

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 9, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 16, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2, 2, 2, 7, 1, 11, 2, 4, 2, 2, 2, 19, 1, 4, 4, 9, 1, 5, 1

Offset: 1

N. J. A. Sloane

From David W. Wilson, Feb 28 2009: (Start)
By a factorization of n we mean a multiset of integers > 1 whose product is n.
For example, 6 is the product of 2 such multisets, {2, 3} and {6}, so a(6) = 2.
Similarly 8 is the product of 3 such multisets, {2, 2, 2}, {2, 4} and {8}, so a(8) = 3.
1 is the product of 1 such multiset, namely the empty multiset {}, whose product is by definition the multiplicative identity 1. Hence a(1) = 1. (End)
a(n) = # { k | A064553(k) = n }. - Reinhard Zumkeller, Sep 21 2001; Benoit Cloitre and N. J. A. Sloane, May 15 2002
Number of members of A025487 with n divisors. - Matthew Vandermast, Jul 12 2004
See sequence A162247 for a list of the factorizations of n and a program for generating the factorizations for any n. - T. D. Noe, Jun 28 2009
So a(n) gives the number of different prime signatures that can be found among the integers that have n divisors. - Michel Marcus, Nov 11 2015
For n > 0, also the number of integer partitions of n with product n, ranked by A301987. For example, the a(12) = 4 partitions are: (12), (6,2,1,1,1,1), (4,3,1,1,1,1,1), (3,2,2,1,1,1,1,1). See also A380218. In general, A379666(m,n) = a(n) for any m >= n. - Gus Wiseman, Feb 07 2025

			1: 1, a(1) = 1
2: 2, a(2) = 1
3: 3, a(3) = 1
4: 4 = 2*2, a(4) = 2
6: 6 = 2*3, a(6) = 2
8: 8 = 2*4 = 2*2*2, a(8) = 3
etc.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 844.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 292-295.
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 198, exercise 9 (in the third edition 2015, p. 296, exercise 211).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
D. Beckwith, Problem 10669, Amer. Math. Monthly 105 (1998), p. 559.
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28. [A second link to the same paper.]
Marc Chamberland, Colin Johnson, Alice Nadeau, and Bingxi Wu, Multiplicative Partitions, Electronic Journal of Combinatorics, 20(2) (2013), #P57.
S. R. Finch, Kalmar's composition constant, Jun 05 2003. [Cached copy, with permission of the author]
Shamik Ghosh, Counting number of factorizations of a natural number, arXiv:0811.3479 [cs.DM], 2008.
R. K. Guy and R. J. Nowakowski, Monthly unsolved problems, 1969-1995, Amer. Math. Monthly, 102 (1995), 921-926.
John F. Hughes and J. O. Shallit, On the Number of Multiplicative Partitions, American Mathematical Monthly 90(7) (1983), 468-471.
Cao Hui-Zhong and Ku Tung-Hsin, On the Enumeration Function of Multiplicative Partitions, Math. Balkanica, Vol. 4 (1990), Fasc. 3-4.
Florian Luca, Anirban Mukhopadhyay and Kotyada Srinivas, On the Oppenheim's "factorisatio numerorum" function, arXiv:0807.0986 [math.NT], 2008.
Pankaj Jyoti Mahanta, On the number of partitions of n whose product of the summands is at most n, arXiv:2010.07353 [math.CO], 2020.
Amarnath Murthy, Generalization of Partition Function (Introducing the Smarandache Factor Partition) [Broken link]
Amarnath Murthy and Charles Ashbacher, Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences, Hexis, Phoenix; USA 2005. See Section 1.4.
Paul Pollack, On the parity of the number of multiplicative partitions and related problems, Proc. Amer. Math. Soc. 140 (2012), 3793-3803.
Marko Riedel, Calculating the multiplicative partition function in Maple with the Polya Enumeration Theorem
Eric Weisstein's World of Mathematics, Unordered Factorization
Wikipedia, Multiplicative Partition Function
Index entries for "core" sequences

A045782 gives the range of a(n).
For records see A033833, A033834.
Cf. A002033, A045778 (strict version), A050322, A050336, A064553, A064554, A064555, A077565, A051731, A005171, A097296, A190938, A216599, A216600, A216601, A216602.
Row sums of A316439 (for n>1).
Cf. A096276 (partial sums).
The additive version is A000041 (integer partitions), strict A000009.
Row sums of A318950.
A002865 counts partitions into parts > 1.
A069016 counts distinct sums of factorizations.
A319000 counts partitions by product and sum, row sums A319916.
A379666 (transpose A380959) counts partitions by sum and product, without 1's A379668, strict A379671.
Cf. A003963, A057567, A057568 (strict A379733), A114324, A301987, A319005, A379734.

a001055 = (map last a066032_tabl !!) . (subtract 1)
-- Reinhard Zumkeller, Oct 01 2012

public class MultiPart {
    public static void main(String[] argV) {
        for (int i=1;i<=100;++i) System.out.println(1+getDivisors(2,i));
    }
    public static int getDivisors(int min,int n) {
        int total = 0;
        for (int i=min;i=i) { ++total; if (n/i>i) total+=getDivisors(i,n/i); }
        return total;
    }
} \\ Scott R. Shannon, Aug 21 2019

with(numtheory):
T := proc(n::integer, m::integer)
        local A, summe, d:
        if isprime(n) then
                if n <= m then
                        return 1;
                end if:
                return 0 ;
        end if:
        A := divisors(n) minus {n, 1}:
        for d in A do
                if d > m then
                        A := A minus {d}:
                end if:
        end do:
        summe := add(T(n/d,d),d=A) ;
        if n <=m then
                summe := summe + 1:
        end if:
        summe ;
end proc:
A001055 := n -> T(n, n):
[seq(A001055(n), n=1..100)]; # Reinhard Zumkeller and Ulrich Schimke (ulrschimke(AT)aol.com)

c[1, r_] := c[1, r]=1; c[n_, r_] := c[n, r] = Module[{ds, i}, ds = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]]; a[n_] := c[n, n]; a/@Range[100] (* c[n, r] is the number of factorizations of n with factors <= r. - Dean Hickerson, Oct 28 2002 *)
T[, 1] = T[1, ] = 1;
T[n_, m_] := T[n, m] = DivisorSum[n, Boole[1 < # <= m] * T[n/#, #]&];
a[n_] := T[n, n];
a /@ Range[100] (* Jean-François Alcover, Jan 03 2020 *)

/* factorizations of n with factors <= m (n,m positive integers) */
fcnt(n,m) = {local(s);s=0;if(n == 1,s=1,fordiv(n,d,if(d > 1 & d <= m,s=s+fcnt(n/d,d))));s}
A001055(n) = fcnt(n,n) \\ Michael B. Porter, Oct 29 2009

\\ code using Dirichlet g.f., based on Somos's code for A007896
{a(n) = my(A, v, w, m);
if(
n<1, 0,
\\ define unit vector v = [1, 0, 0, ...] of length n
v = vector(n, k, k==1);
for(k=2, n,
m = #digits(n, k) - 1;
\\ expand 1/(1-x)^k out far enough
A = (1 - x)^ -1 + x * O(x^m);
\\ w = zero vector of length n
w = vector(n);
\\ convert A to a vector
for(i=0, m, w[k^i] = polcoeff(A, i));
\\ build the answer
v = dirmul(v, w)
);
v[n]
)
};
\\ produce the sequence
vector(100,n,a(n)) \\ N. J. A. Sloane, May 26 2014

v=vector(100, k, k==1); for(n=2, #v, v+=dirmul(v, vector(#v, k, (k>1) && n^valuation(k,n)==k)) ); v \\ Max Alekseyev, Jul 16 2014

from sympy import divisors, isprime
def T(n, m):
    if isprime(n): return 1 if n<=m else 0
    A=filter(lambda d: d<=m, divisors(n)[1:-1])
    s=sum(T(n//d, d) for d in A)
    return s + 1 if n<=m else s
def a(n): return T(n, n)
print([a(n) for n in range(1, 106)]) # Indranil Ghosh, Aug 19 2017

The asymptotic behavior of this sequence was studied by Canfield, Erdős & Pomerance and Luca, Mukhopadhyay, & Srinivas. - Jonathan Vos Post, Jul 07 2008
Dirichlet g.f.: Product_{k>=2} 1/(1 - 1/k^s).
If n = p^k for a prime p, a(n) = partitions(k) = A000041(k).
Since the sequence a(n) is the right diagonal of A066032, the given recursive formula for A066032 applies (see Maple program). - Reinhard Zumkeller and Ulrich Schimke (ulrschimke(AT)aol.com)
a(A002110(n)) = A000110(n).
a(p^k*q^k) = A002774(k) if p and q are distinct primes. - R. J. Mathar, Jun 06 2024
a(n) = A028422(n) + 1. - Gus Wiseman, Feb 07 2025

Incorrect assertion about asymptotic behavior deleted by N. J. A. Sloane, Jun 08 2009

A072233 Square array T(n,k) read by antidiagonals giving number of ways to distribute n indistinguishable objects in k indistinguishable containers; containers may be left empty.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 4, 5, 5, 3, 2, 1, 1, 0, 1, 4, 7, 6, 5, 3, 2, 1, 1, 0, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 0, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 0, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 0, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 0

Martin Wohlgemuth (mail(AT)matroid.com), Jul 05 2002

Regarded as a triangular table, this is another version of the number of partitions of n into k parts, A008284. - Franklin T. Adams-Watters, Dec 18 2006
From Gus Wiseman, Feb 10 2021: (Start)
T(n,k) is also the number of partitions of n with greatest part k, if we assume the greatest part of an empty partition to be 0. Row n = 9 counts the following partitions:
  111111111  22221     333      432     54     63    72   81  9
             222111    3222     441     522    621   711
             2211111   3321     4221    531    6111
             21111111  32211    4311    5211
                       33111    42111   51111
                       321111   411111
                       3111111
(End)

			Table begins (upper left corner = T(0,0)):
1 1 1  1  1  1  1  1  1 ...
0 1 1  1  1  1  1  1  1 ...
0 1 2  2  2  2  2  2  2 ...
0 1 2  3  3  3  3  3  3 ...
0 1 3  4  5  5  5  5  5 ...
0 1 3  5  6  7  7  7  7 ...
0 1 4  7  9 10 11 11 11 ...
0 1 4  8 11 13 14 15 15 ...
0 1 5 10 15 18 20 21 22 ...
There is 1 way to distribute 0 objects into k containers: T(0, k) = 1. The different ways for n=4, k=3 are: (oooo)()(), (ooo)(o)(), (oo)(oo)(), (oo)(o)(o), so T(4, 3) = 4.
From _Wolfdieter Lang_, Dec 03 2012 (Start)
The triangle a(n,k) = T(n-k,k) begins:
n\k  0  1  2  3  4  5  6  7  8  9 10 ...
00   1
01   0  1
02   0  1  1
03   0  1  1  1
04   0  1  2  1  1
05   0  1  2  2  1  1
06   0  1  3  3  2  1  1
07   0  1  3  4  3  2  1  1
08   0  1  4  5  5  3  2  1  1
09   0  1  4  7  6  5  3  2  1  1
10   0  1  5  8  9  7  5  3  2  1  1
...
Row n=5 is, for k=1..5, [1,2,2,1,1] which gives the number of partitions of n=5 with k parts. See A008284 and the Franklin T. Adams-Watters comment above. (End)
From _Gus Wiseman_, Feb 10 2021: (Start)
Row n = 9 counts the following partitions:
  9  54  333  3222  22221  222111  2211111  21111111  111111111
     63  432  3321  32211  321111  3111111
     72  441  4221  33111  411111
     81  522  4311  42111
         531  5211  51111
         621  6111
         711
(End)

Robert G. Wilson v, Table of n, a(n) for n = 0..10010
Combinatorial Object Server, Information on Numerical Partitions
FindStat - Combinatorial Statistic Finder, The length of the partition.

Sum of antidiagonal entries T(n, k) with n+k=m equals A000041(m).
Alternating row sums are A081362.
Cf. A008284.
The version for factorizations is A316439.
The version for set partitions is A048993/A080510.
The version for strict partitions is A008289/A059607.
A047993 counts balanced partitions, ranked by A106529.
A063995/A105806 count partitions by Dyson rank.
Cf. A006141, A064174, A096401, A117409, A168659, A215366.

Flatten[Table[Length[IntegerPartitions[n, {k}]], {n, 0, 20}, {k, 0, n}]] (* Emanuele Munarini, Feb 24 2014 *)

from sage.combinat.partition import number_of_partitions_length
[[number_of_partitions_length(n, k) for k in (0..n)] for n in (0..10)] # Peter Luschny, Aug 01 2015

T(0, k) = 1, T(n, 0) = 0 (n>0), T(1, k) = 1 (k>0), T(n, 1) = 1 (n>0), T(n, k) = 0 for n < 0, T(n, k) = Sum[ T(n-k+i, k-i), i=0...k-1] Or, T(n, 1) = T(n, n) = 1, T(n, k) = 0 (k>n), T(n, k) = T(n-1, k-1) + T(n-k, k).
G.f. Product_{j=0..infinity} 1/(1-xy^j). Regarded as a triangular array, g.f. Product_{j=1..infinity} 1/(1-xy^j). - Franklin T. Adams-Watters, Dec 18 2006
O.g.f. of column No. k of the triangle a(n,k) is x^k/product(1-x^j,j=1..k), k>=0 (the undefined product for k=0 is put to 1). - Wolfdieter Lang, Dec 03 2012

Corrected by Franklin T. Adams-Watters, Dec 18 2006

A162247 Irregular triangle in which row n lists all factorizations of n, sorted by the number of factors in each factorization.

Original entry on oeis.org

1, 2, 3, 4, 2, 2, 5, 6, 2, 3, 7, 8, 2, 4, 2, 2, 2, 9, 3, 3, 10, 2, 5, 11, 12, 2, 6, 3, 4, 2, 2, 3, 13, 14, 2, 7, 15, 3, 5, 16, 2, 8, 4, 4, 2, 2, 4, 2, 2, 2, 2, 17, 18, 2, 9, 3, 6, 2, 3, 3, 19, 20, 2, 10, 4, 5, 2, 2, 5, 21, 3, 7, 22, 2, 11, 23, 24, 2, 12, 3, 8, 4, 6, 2, 2, 6, 2, 3, 4, 2, 2, 2, 3, 25, 5, 5

Offset: 1

T. D. Noe, Jun 28 2009

Row n begins with n because it is a factorization of length 1. In each factorization, the factors are in nondecreasing order. This sequence is A056472 with the factorizations in a different order. Sequence A001055(n) gives the number of factorizations of n; A066637(n) gives the number of numbers in row n. In the Mathematica program, the function f returns a list of the factorizations of n.

These factorizations are useful in determining the forms of numbers that have a given number of divisors. For example, to find the forms of numbers that have 12 divisors, we look at the four factorizations of 12 (12, 2*6, 3*4, 2*2*3), subtract 1 from each factor, and find the forms to be p^11, p q^5, p^2 q^3, and p q r^2, where p, q, and r are prime numbers.

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

See A001055.

T. D. Noe, Rows n=1..1000 of triangle, flattened
R. J. Mathar, Factorizations of n=1..1100

Cf. A001055, A056472, A066637, A316439.

import Data.List (sortBy)
import Data.Ord (comparing)
a162247 n k = a162247_tabl !! (n-1) !! (k-1)
a162247_row n = a162247_tabl !! (n-1)
a162247_tabl = map (concat . sortBy (comparing length)) $ tail fss where
   fss = [] : map fact [1..] where
         fact x = [x] : [d : fs | d <- [2..x], let (x',r) = divMod x d,
                                  r == 0, fs <- fss !! x', d <= head fs]
-- Reinhard Zumkeller, Jan 08 2013

g[lst_,p_] := Module[{t,i,j}, Union[Flatten[Table[t=lst[[i]]; t[[j]]=p*t[[j]]; Sort[t], {i,Length[lst]}, {j,Length[lst[[i]]]}], 1], Table[Sort[Append[lst[[i]],p]], {i,Length[lst]}]]]; f[n_] := Module[{i,j,p,e,lst={{}}}, {p,e}=Transpose[FactorInteger[n]]; Do[lst=g[lst,p[[i]]], {i,Length[p]}, {j,e[[i]]}]; lst]; Flatten[Table[f[n], {n,25}]]

A339890 Number of odd-length factorizations of n into factors > 1.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Dec 28 2020

nonn

			The a(n) factorizations for n = 24, 48, 60, 72, 96, 120:
  24      48          60       72          96          120
  2*2*6   2*3*8       2*5*6    2*4*9       2*6*8       3*5*8
  2*3*4   2*4*6       3*4*5    2*6*6       3*4*8       4*5*6
          3*4*4       2*2*15   3*3*8       4*4*6       2*2*30
          2*2*12      2*3*10   3*4*6       2*2*24      2*3*20
          2*2*2*2*3            2*2*18      2*3*16      2*4*15
                               2*3*12      2*4*12      2*5*12
                               2*2*2*3*3   2*2*2*2*6   2*6*10
                                           2*2*2*3*4   3*4*10
                                                       2*2*2*3*5

Alois P. Heinz, Table of n, a(n) for n = 1..20000

The case of set partitions (or n squarefree) is A024429.
The case of partitions (or prime powers) is A027193.
The ordered version is A174726 (even: A174725).
The remaining (even-length) factorizations are counted by A339846.
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.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A316439 counts factorizations by product and length.
A340101 counts factorizations into odd factors.
A340102 counts odd-length factorizations into odd factors.
Cf. A000700, A002033, A027187, A028260, A074206, A078408, A236914, A320732.

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<2, 0, g(n$2, 1)):
seq(a(n), n=1..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[#]&]],{n,100}]

a(n) + A339846(n) = A001055(n).

A339846 Number of even-length factorizations of n into factors > 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 3, 0, 2, 0, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 3, 0, 3, 1, 1, 1, 5, 0, 1, 1, 4, 0, 3, 0, 2, 2, 1, 0, 6, 1, 2, 1, 2, 0, 4, 1, 4, 1, 1, 0, 6, 0, 1, 2, 6, 1, 3, 0, 2, 1, 3, 0, 8, 0, 1, 2, 2, 1, 3, 0, 6, 3, 1, 0, 6, 1, 1, 1, 4, 0, 6, 1, 2, 1, 1, 1, 10, 0, 2, 2, 5, 0, 3, 0, 4, 3

Offset: 1

Gus Wiseman, Dec 28 2020

nonn

			The a(n) factorizations for n = 12, 16, 24, 36, 48, 72, 96, 120:
  2*6  2*8      3*8      4*9      6*8      8*9      2*48         2*60
  3*4  4*4      4*6      6*6      2*24     2*36     3*32         3*40
       2*2*2*2  2*12     2*18     3*16     3*24     4*24         4*30
                2*2*2*3  3*12     4*12     4*18     6*16         5*24
                         2*2*3*3  2*2*2*6  6*12     8*12         6*20
                                  2*2*3*4  2*2*2*9  2*2*3*8      8*15
                                           2*2*3*6  2*2*4*6      10*12
                                           2*3*3*4  2*3*4*4      2*2*5*6
                                                    2*2*2*12     2*3*4*5
                                                    2*2*2*2*2*3  2*2*2*15
                                                                 2*2*3*10

The case of set partitions (or n squarefree) is A024430.
The case of partitions (or prime powers) is A027187.
The ordered version is A174725, odd: A174726.
The odd-length factorizations are counted by A339890.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A316439 counts factorizations by product and length.
A340102 counts odd-length factorizations into odd factors.
Cf. A002033, A007716, A027193, A050320, A058695, A074206, A236913, A320655, A320656, A320732.

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=1, 1, g(n$2, 0)):
seq(a(n), n=1..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],EvenQ@Length[#]&]],{n,100}]

A339846(n, m=n, e=1) = if(1==n, e, sumdiv(n, d, if((d>1)&&(d<=m), A339846(n/d, d, 1-e)))); \\ Antti Karttunen, Oct 22 2023

a(n) + A339890(n) = A001055(n).

Data section extended up to a(105) by Antti Karttunen, Oct 22 2023

A340653 Number of balanced factorizations of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

A factorization into factors > 1 is balanced if it is empty or its length is equal to its maximum Omega (A001222).

			The balanced factorizations for n = 120, 144, 192, 288, 432, 768:
  3*5*8    2*8*9    3*8*8      4*8*9      6*8*9      8*8*12
  2*2*30   3*6*8    4*6*8      6*6*8      2*8*27     2*2*8*24
  2*3*20   2*4*18   2*8*12     2*8*18     3*8*18     2*3*8*16
  2*5*12   2*6*12   4*4*12     3*8*12     4*4*27     2*4*4*24
           3*4*12   2*2*2*24   4*4*18     4*6*18     2*4*6*16
                    2*2*3*16   4*6*12     4*9*12     3*4*4*16
                               2*12*12    6*6*12     2*2*12*16
                               2*2*2*36   2*12*18    2*2*2*2*48
                               2*2*3*24   3*12*12    2*2*2*3*32
                               2*3*3*16   2*2*2*54
                                          2*2*3*36
                                          2*3*3*24
                                          3*3*3*16

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Positions of zeros are A001358.
Positions of nonzero terms are A100959.
The co-balanced version is A340596.
Taking maximum factor instead of maximum Omega gives A340599.
The cross-balanced version is A340654.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A320655 counts factorizations into semiprimes.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340597 have an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
- A340656 have no twice-balanced factorizations.
- A340657 have a twice-balanced factorization.
Cf. A003963, A117409, A303975, A320656, A339846, A339890, A340608.

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],#=={}||Length[#]==Max[PrimeOmega/@#]&]],{n,100}]

A340653(n, m=n, mbo=0, e=0) = if(1==n, mbo==e, sumdiv(n, d, if((d>1)&&(d<=m), A340653(n/d, d, max(mbo,bigomega(d)), 1+e)))); \\ Antti Karttunen, Oct 22 2023

Data section extended up to a(120) by Antti Karttunen, Oct 22 2023

A340596 Number of co-balanced factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 2, 1, 1, 1, 5, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 8

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

We define a factorization of n into factors > 1 to be co-balanced if it has exactly A001221(n) factors.

			The a(n) co-balanced factorizations for n = 12, 24, 36, 72, 120, 144, 180:
  2*6    3*8     4*9     8*9     3*5*8     2*72     4*5*9
  3*4    4*6     6*6     2*36    4*5*6     3*48     5*6*6
         2*12    2*18    3*24    2*2*30    4*36     2*2*45
                 3*12    4*18    2*3*20    6*24     2*3*30
                         6*12    2*4*15    8*18     2*5*18
                                 2*5*12    9*16     2*6*15
                                 2*6*10    12*12    2*9*10
                                 3*4*10             3*3*20
                                                    3*4*15
                                                    3*5*12
                                                    3*6*10

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

Positions of terms > 1 are A126706.
Positions of 1's are A303554.
The version for unlabeled multiset partitions is A319616.
The alt-balanced version is A340599.
The balanced version is A340653.
The cross-balanced version is A340654.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340597 lists numbers with an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A003963, A006141, A050320, A112798, A117409, A324518, A339846, A339890, A340607, A340656, A340657.

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],Length[#]==PrimeNu[n]&]],{n,100}]

A340596(n, m=n, om=omega(n)) = if(1==n,(0==om), sumdiv(n, d, if((d>1)&&(d<=m), A340596(n/d, d, om-1)))); \\ Antti Karttunen, Jun 10 2024

Data section extended up to a(120) by Antti Karttunen, Jun 10 2024

A379666 Array read by antidiagonals downward where A(n,k) is the number of integer partitions of n with product k.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 01 2025

Counts finite multisets of positive integers by sum and product.

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  1   0   0   0   0   0   0   0   0   0   0   0
   n=2:  1   1   0   0   0   0   0   0   0   0   0   0
   n=3:  1   1   1   0   0   0   0   0   0   0   0   0
   n=4:  1   1   1   2   0   0   0   0   0   0   0   0
   n=5:  1   1   1   2   1   1   0   0   0   0   0   0
   n=6:  1   1   1   2   1   2   0   2   1   0   0   0
   n=7:  1   1   1   2   1   2   1   2   1   1   0   2
   n=8:  1   1   1   2   1   2   1   3   1   1   0   3
   n=9:  1   1   1   2   1   2   1   3   2   1   0   3
  n=10:  1   1   1   2   1   2   1   3   2   2   0   3
  n=11:  1   1   1   2   1   2   1   3   2   2   1   3
  n=12:  1   1   1   2   1   2   1   3   2   2   1   4
For example, the A(9,12) = 3 partitions are: (6,2,1), (4,3,1,1), (3,2,2,1,1).
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 1
   n+k=3: 0 0 1
   n+k=4: 0 0 1 1
   n+k=5: 0 0 0 1 1
   n+k=6: 0 0 0 1 1 1
   n+k=7: 0 0 0 0 1 1 1
   n+k=8: 0 0 0 0 2 1 1 1
   n+k=9: 0 0 0 0 0 2 1 1 1
  n+k=10: 0 0 0 0 0 1 2 1 1 1
  n+k=11: 0 0 0 0 0 1 1 2 1 1 1
  n+k=12: 0 0 0 0 0 0 2 1 2 1 1 1
  n+k=13: 0 0 0 0 0 0 0 2 1 2 1 1 1
  n+k=14: 0 0 0 0 0 0 2 1 2 1 2 1 1 1
  n+k=15: 0 0 0 0 0 0 1 2 1 2 1 2 1 1 1
  n+k=16: 0 0 0 0 0 0 0 1 3 1 2 1 2 1 1 1
For example, antidiagonal n+k=10 counts the following partitions:
  n=5: (5)
  n=6: (411), (2211)
  n=7: (31111)
  n=8: (2111111)
  n=9: (111111111)
so the 10th antidiagonal is: (0,0,0,0,0,1,2,1,1,1).

Row sums are A000041 = partitions of n, strict A000009, no ones A002865.
Diagonal A(n,n) is A001055(n) = factorizations of n, strict A045778.
Antidiagonal sums are A379667.
The case without ones is A379668, antidiagonal sums A379669 (zeros A379670).
The strict case is A379671, antidiagonal sums A379672.
The strict case without ones is A379678, antidiagonal sums A379679 (zeros A379680).
A316439 counts factorizations by length, partitions A008284.
A326622 counts factorizations with integer mean, strict A328966.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A003963, A028422, A069016, A096765, A318950, A319000, A319916, A319057, A325036, A325041, A325042, A326152.

nn=12;
tt=Table[Length[Select[IntegerPartitions[n],Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

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

A379671 Array read by antidiagonals downward where A(n,k) is the number of finite sets of positive integers with sum n and product k.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 01 2025

Counts finite sets of positive integers by sum and product.

			Array begins:
        k=1 k=2 k=3 k=4 k=5 k=6 k=7 k=8 k=9 k10 k11 k12
        -----------------------------------------------
   n=0:  1   0   0   0   0   0   0   0   0   0   0   0
   n=1:  1   0   0   0   0   0   0   0   0   0   0   0
   n=2:  0   1   0   0   0   0   0   0   0   0   0   0
   n=3:  0   1   1   0   0   0   0   0   0   0   0   0
   n=4:  0   0   1   1   0   0   0   0   0   0   0   0
   n=5:  0   0   0   1   1   1   0   0   0   0   0   0
   n=6:  0   0   0   0   1   2   0   1   0   0   0   0
   n=7:  0   0   0   0   0   1   1   1   0   1   0   1
   n=8:  0   0   0   0   0   0   1   1   0   1   0   2
   n=9:  0   0   0   0   0   0   0   1   1   0   0   1
  n=10:  0   0   0   0   0   0   0   0   1   1   0   0
  n=11:  0   0   0   0   0   0   0   0   0   1   1   0
  n=12:  0   0   0   0   0   0   0   0   0   0   1   1
The A(8,12) = 2 sets are: {2,6}, {1,3,4}.
The A(14,40) = 2 sets are: {4,10}, {1,5,8}.
Antidiagonals begin:
   n+k=1: 1
   n+k=2: 0 1
   n+k=3: 0 0 0
   n+k=4: 0 0 1 0
   n+k=5: 0 0 0 1 0
   n+k=6: 0 0 0 1 0 0
   n+k=7: 0 0 0 0 1 0 0
   n+k=8: 0 0 0 0 1 0 0 0
   n+k=9: 0 0 0 0 0 1 0 0 0
  n+k=10: 0 0 0 0 0 1 0 0 0 0
  n+k=11: 0 0 0 0 0 1 1 0 0 0 0
  n+k=12: 0 0 0 0 0 0 2 0 0 0 0 0
  n+k=13: 0 0 0 0 0 0 0 1 0 0 0 0 0
  n+k=14: 0 0 0 0 0 0 1 1 0 0 0 0 0 0
  n+k=15: 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0
  n+k=16: 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
For example, antidiagonal n+k=11 counts the following sets:
  n=5: {2,3}
  n=6: {1,5}
so the 11th antidiagonal is: (0,0,0,0,0,1,1,0,0,0,0).

Row sums are A000009 = strict partitions, non-strict A000041.
Column sums are 2*A045778 where A045778 = strict factorizations, non-strict A001055.
Antidiagonal sums are A379672, non-strict A379667 (zeros A379670).
Without ones we have A379678, antidiagonal sums A379679 (zeros A379680).
The non-strict version is A379666, without ones A379668.
A316439 counts factorizations by length, partitions A008284.
A326622 counts factorizations with integer mean, strict A328966.
Counting and ranking multisets by comparing sum and product:
- same: A001055, ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A379733
- greater than: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less than: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721, see A025147
- different: A379736, ranks A379722, see A111133
Cf. A002865, A003963, A028422, A069016, A318950, A319000, A319916, A319057, A325036, A325041, A325042, A326152.

nn=12;
tt=Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Times@@#==k&]],{n,0,nn},{k,1,nn}] (* array *)
tr=Table[tt[[j,i-j]],{i,2,nn},{j,i-1}] (* antidiagonals *)
Join@@tr (* sequence *)

cp's OEIS Frontend

A001055 The multiplicative partition function: number of ways of factoring n with all factors greater than 1 (a(1) = 1 by convention).

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A072233 Square array T(n,k) read by antidiagonals giving number of ways to distribute n indistinguishable objects in k indistinguishable containers; containers may be left empty.

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A162247 Irregular triangle in which row n lists all factorizations of n, sorted by the number of factors in each factorization.

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A339890 Number of odd-length factorizations of n into factors > 1.

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A339846 Number of even-length factorizations of n into factors > 1.

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A340653 Number of balanced factorizations of n.

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