A050336 -id:A050336 - 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

A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

Original entry on oeis.org

1, 1, 3, 6, 14, 27, 58, 111, 223, 424, 817, 1527, 2870, 5279, 9710, 17622, 31877, 57100, 101887, 180406, 318106, 557453, 972796, 1688797, 2920123, 5026410, 8619551, 14722230, 25057499, 42494975, 71832114, 121024876, 203286806, 340435588, 568496753, 946695386

Offset: 0

N. J. A. Sloane

a(n) = number of partitions of n, when for each k there are p(k) different copies of part k. E.g., let the parts be 1, 2a, 2b, 3a, 3b, 3c, 4a, 4b, 4c, 4d, 4e, ... Then the a(4) = 14 partitions of 4 are: 4 = 4a = 4b = ... = 4e = 3a+1 = 3b+1 = 3c+1 = 2a+2a = 2a+2b = 2b+2b = 2a+1 = 2b+1 = 1+1+1+1.
Equivalently (Cayley), a(n) = number of 2-dimensional partitions of n. E.g., for n = 4 we have:
  4   31   3   22   2   211   21   2    2   1111   111   11    11   1
           1        2         1    11   1          1     11    1    1
                                        1                      1    1
                                                                    1
Also total number of different species of singularity for conjugate functions with n letters (Sylvester).
According to [Belmans], this sequence gives "[t]he number of Segre symbols for the intersection of two quadrics in a fixed dimension". - Eric M. Schmidt, Sep 02 2017
From Gus Wiseman, Jul 30 2022: (Start)
Also the number of non-isomorphic multiset partitions of weight n with all constant blocks. The strict case is A089259. For example, non-isomorphic representatives of the a(1) = 1 through a(3) = 6 multiset partitions are:
  {{1}}  {{1,1}}    {{1,1,1}}
         {{1},{1}}  {{1},{1,1}}
         {{1},{2}}  {{1},{2,2}}
                    {{1},{1},{1}}
                    {{1},{2},{2}}
                    {{1},{2},{3}}
A000688 counts factorizations into prime powers.
A007716 counts non-isomorphic multiset partitions by weight.
A279784 counts twice-partitions of type PPR, factorizations A295935.
Constant partitions are ranked by prime-powers: A000961, A023894, A054685, A246655, A355743.
(End)

			G.f. = 1 + x + 3*x^2 + 6*x^3 + 15*x^4 + 28*x^5 + 66*x^6 + 122*x^7 + ...
a(3) = 6 because we have (111) = (111) = (11)(1) = (1)(1)(1), (12) = (12) = (1)(2), (3) = (3).
The a(4)=14 multiset partitions whose total sum of parts is 4 are:
((4)),
((13)), ((1)(3)),
((22)), ((2)(2)),
((112)), ((1)(12)), ((2)(11)), ((1)(1)(2)),
((1111)), ((1)(111)), ((11)(11)), ((1)(1)(11)), ((1)(1)(1)(1)). - _Gus Wiseman_, Dec 19 2016

A. Cayley, Recherches sur les matrices dont les termes sont des fonctions linéaires d'une seule indéterminée, J. Reine angew. Math., 50 (1855), 313-317; Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 2, p. 219.
V. A. Liskovets, Counting rooted initially connected directed graphs. Vesci Akad. Nauk. BSSR, ser. fiz.-mat., No 5, 23-32 (1969), MR44 #3927.
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).
J. J. Sylvester, An Enumeration of the Contacts of Lines and Surfaces of the Second Order, Phil. Mag. 1 (1851), 119-140. Reprinted in Collected Papers, Vol. 1. See p. 239, where one finds a(n)-2, but with errors.
J. J. Sylvester, Note on the 'Enumeration of the Contacts of Lines and Surfaces of the Second Order', Phil. Mag., Vol. VII (1854), pp. 331-334. Reprinted in Collected Papers, Vol. 2, pp. 30-33.

Reinhard Zumkeller, Table of n, a(n) for n = 0..5000 (first 500 terms from T. D. Noe)
Pieter Belmans, Segre symbols, 2016.
Philip Boalch, Counting the fission trees and nonabelian Hodge graphs, arXiv:2410.23358 [math.AG], 2024. See pp. 10, 16.
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 148
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).
L. Kaylor and D. Offner, Counting matrices over a finite field with all eigenvalues in the field, Involve, a Journal of Mathematics, Vol. 7 (2014), No. 5, 627-645. [DOI]
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious pairs, 2014.
M. Kozek, F. Luca, P. Pollack, and C. Pomerance, Harmonious numbers, IJNT, to appear.
XiKun Li, JunLi Li, Bin Liu and CongFeng Qiao, The parametric symmetry and numbers of the entangled class of 2 × M × N system, Science China Physics, Mechanics & Astronomy, Volume 54, Number 8, 1471-1475, DOI: 10.1007/s11433-011-4395-9.
Jessie Pitsillides, Segre Characteristic Equivalence, arXiv:2506.12065 [math.GM], 2025.
Paul Pollack and Carl Pomerance, Some problems of Erdős on the sum-of-divisors function, For Richard Guy on his 99th birthday: May his sequence be unbounded, Trans. Amer. Math. Soc. Ser. B, Vol. 3 (2016), pp. 1-26; Errata.
N. J. A. Sloane, Transforms.
N. J. A. Sloane and Thomas Wieder, The Number of Hierarchical Orderings, arXiv:math/0307064 [math.CO], 2003; Order 21 (2004), 83-89.
J. J. Sylvester, The collected mathematical papers of James Joseph Sylvester, vol. 2, vol. 3, vol. 4.
Index entries for sequences related to rooted trees

Related to A001383 via generating function.
The multiplicative version (factorizations) is A050336.
The ordered version (sequences of partitions) is A055887.
Row-sums of A061260.
Main diagonal of A055885.
We have A271619(n) <= a(n) <= A063834(n).
Column k=3 of A290353.
The strict case is A316980.
Cf. A000041, A061259, A006171, A061255, A061256, A061257, A089292, A000219.
Cf. A089300.
Cf. A001055, A072233, A112798, A275024.

Following Vladeta Jovovic:
a001970 n = a001970_list !! (n-1)
a001970_list = 1 : f 1 [1] where
   f x ys = y : f (x + 1) (y : ys) where
            y = sum (zipWith (*) ys a061259_list) `div` x
-- Reinhard Zumkeller, Oct 31 2015

with(combstruct); SetSetSetU := [T, {T=Set(S), S=Set(U,card >= 1), U=Set(Z,card >=1)},unlabeled];
# second Maple program:
with(numtheory): with(combinat):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      numbpart(d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 19 2016

m = 32; f[x_] = Product[1/(1-x^k)^PartitionsP[k], {k, 1, m}]; CoefficientList[ Series[f[x], {x, 0, m-1}], x] (* Jean-François Alcover, Jul 19 2011, after g.f. *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, 1 - numbpart(k) * x^k + x * O(x^n)), n))}; /* Michael Somos, Dec 20 2016 */

from sympy.core.cache import cacheit
from sympy import npartitions, divisors
@cacheit
def a(n): return 1 if n == 0 else sum([sum([d*npartitions(d) for d in divisors(j)])*a(n - j) for j in range(1, n + 1)]) / n
[a(n) for n in range(51)]  # Indranil Ghosh, Aug 19 2017, after Maple code
# (Sage) # uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, 1)
a = EulerTransform(EulerTransform(b))
print([a(n) for n in range(36)]) # Peter Luschny, Nov 17 2022

G.f.: Product_{k >= 1} 1/(1-x^k)^p(k), where p(k) = number of partitions of k = A000041. [Cayley]
a(n) = (1/n)*Sum_{k = 1..n} a(n-k)*b(k), n > 1, a(0) = 1, b(k) = Sum_{d|k} d*numbpart(d), where numbpart(d) = number of partitions of d, cf. A061259. - Vladeta Jovovic, Apr 21 2001
Logarithmic derivative yields A061259 (equivalent to above formula from Vladeta Jovovic). - Paul D. Hanna, Sep 05 2012
a(n) = Sum_{k=1..A000041(n)} A001055(A215366(n,k)) = number of factorizations of Heinz numbers of integer partitions of n. - Gus Wiseman, Dec 19 2016
a(n) = |{m>=1 : n = Sum_{k=1..A001222(m)} A056239(A112798(m,k)+1)}| = number of normalized twice-prime-factored multiset partitions (see A275024) whose total sum of parts is n. - Gus Wiseman, Dec 19 2016

Additional comments from Valery A. Liskovets

Sylvester references from Barry Cipra, Oct 07 2003

A281113 Number of twice-factorizations of n. Number of ways to choose a postpositive factorization of each part of a postpositive factorization of n.

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 6, 3, 3, 1, 9, 1, 3, 3, 15, 1, 9, 1, 9, 3, 3, 1, 23, 3, 3, 6, 9, 1, 12, 1, 28, 3, 3, 3, 32, 1, 3, 3, 23, 1, 12, 1, 9, 9, 3, 1, 58, 3, 9, 3, 9, 1, 23, 3, 23, 3, 3, 1, 41, 1, 3, 9, 66, 3, 12, 1, 9, 3, 12, 1, 84, 1, 3, 9, 9, 3, 12, 1, 58, 15, 3

Offset: 2

Gus Wiseman, Jan 14 2017

nonn

A postpositive number is a positive integer other than 1. A postpositive factorization of n is a finite orderless sequence of postpositive numbers whose product is n.

			The a(20)=9 twice-factorizations are: ((20)), ((2*10)), ((4*5)), ((2*2*5)), ((2)*(10)), ((2)*(2*5)), ((4)*(5)), ((2*2)*(5)), ((2)*(2)*(5)).
Twice-factorizations of 32 organized by composite:
((2)(2)(2)(2)(2)) ((2)(2)(2)(2 2)) ((2)(2)(2 2 2)) ((2)(2 2)(2 2)) ((2)(2 2 2 2)) ((2 2)(2 2 2)) ((2 2 2 2 2))
((2)(2)(2)(4))    ((2)(2)(2 4))    ((2)(2 2)(4))   ((2)(4)(2 2))   ((2)(2 2 4))   ((2 2)(2 4))   ((4)(2 2 2))  ((2 2 2 4))
((2)(2)(8))       ((2)(2 8))       ((2 2)(8))      ((2 2 8))
((2)(4)(4))       ((2)(4 4))       ((4)(2 4))      ((2 4 4))
((2)(16))         ((2 16))
((4)(8))          ((4 8))
((32)).
Twice-factorizations of 32 organized by domain:
((2)(2)(2)(2)(2))
((2)(2)(2)(2 2)) ((2)(2)(2)(4))
((2)(2)(2 2 2))  ((2)(2)(2 4)) ((2)(2)(8))
((2)(2 2)(2 2))  ((2)(2 2)(4)) ((2)(4)(2 2)) ((2)(4)(4))
((2)(2 2 2 2))   ((2)(2 2 4))  ((2)(2 8))    ((2)(4 4))   ((2)(16))
((2 2)(2 2 2))   ((2 2)(2 4))  ((2 2)(8))    ((4)(2 2 2)) ((4)(2 4)) ((4)(8))
((2 2 2 2 2))    ((2 2 2 4))   ((2 2 8))     ((2 4 4))    ((2 16))   ((4 8)) ((32)).

Michael De Vlieger, Table of n, a(n) for n = 2..30030
Michael De Vlieger, Indices of records in A281113.

Cf. A001055(n) = number of factorizations of n, A050336(n) = number of orderless twice-factorizations of n, A162247(n) = factors of factorizations of n, A063834(n) = a(p^(n-1)), A007716, A269134, A281116.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
twicefacs[n_]:=Join@@Tuples/@Map[postfacs,postfacs[n],{2}];
Table[Length[twicefacs[n]],{n,2,24}]

A292504 Number of orderless tree-factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 4, 2, 2, 1, 6, 1, 2, 2, 11, 1, 6, 1, 6, 2, 2, 1, 20, 2, 2, 4, 6, 1, 8, 1, 30, 2, 2, 2, 27, 1, 2, 2, 20, 1, 8, 1, 6, 6, 2, 1, 74, 2, 6, 2, 6, 1, 20, 2, 20, 2, 2, 1, 38, 1, 2, 6, 96, 2, 8, 1, 6, 2, 8, 1, 114, 1, 2, 6, 6, 2, 8, 1, 74, 11, 2, 1

Offset: 1

Gus Wiseman, Sep 17 2017

nonn

A factorization of n is a finite multiset of positive integers greater than 1 with product n. An orderless tree-factorization of n is either (case 1) the number n itself or (case 2) a finite multiset of two or more orderless tree-factorizations, one of each factor in a factorization of n.

a(n) depends only on the prime signature of n. - Andrew Howroyd, Nov 18 2018

			The a(16)=11 orderless tree-factorizations are: 16, (28), (2(24)), (2(2(22))), (2(222)), (44), (4(22)), ((22)(22)), (224), (22(22)), (2222).

Michael De Vlieger, Table of n, a(n) for n = 1..16384
Michael De Vlieger, Records and indices of records.

Cf. A000311, A001055, A050336, A141268, A281118, A292505.

postfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[postfacs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
oltfacs[n_]:=If[n<=1,{{}},Prepend[Union@@Function[q,Sort/@Tuples[oltfacs/@q]]/@DeleteCases[postfacs[n],{n}],n]];
Table[Length[oltfacs[n]],{n,83}]

seq(n)={my(v=vector(n), w=vector(n)); w[1]=v[1]=1; for(k=2, n, w[k]=v[k]+1; forstep(j=n\k*k, k, -k, my(i=j, e=0); while(i%k==0, i/=k; e++; v[j] += binomial(e+w[k]-1, e)*v[i]))); w} \\ Andrew Howroyd, Nov 18 2018

a(p^n) = A141268(n) for prime p. - Andrew Howroyd, Nov 18 2018

A066739 Number of representations of n as a sum of products of positive integers. 1 is not allowed as a factor, unless it is the only factor. Representations which differ only in the order of terms or factors are considered equivalent.

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 14, 19, 32, 44, 67, 91, 139, 186, 269, 362, 518, 687, 960, 1267, 1747, 2294, 3106, 4052, 5449, 7063, 9365, 12092, 15914, 20422, 26639, 34029, 44091, 56076, 72110, 91306, 116808, 147272, 187224, 235201, 297594, 372390, 468844, 584644, 732942

Offset: 0

Naohiro Nomoto, Jan 16 2002

			For n=5, 5 = 4+1 = 2*2+1 = 3+2 = 3+1+1 = 2+2+1 = 2+1+1+1 = 1+1+1+1+1, so a(5) = 8.
For n=8, 8 = 4*2 = 2*2*2 = ... = 4+4 = 2*2+4 = 2*2+2*2 = ...; note that there are 3 ways to factor the terms of 4+4. In general, if a partition contains a number k exactly r times, then the number of ways to factor the k's is the binomial coefficient C(A001055(k)+r-1,r).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms

Cf. A000041, A001055.
Cf. A066815, A066816, A066806.
Cf. A001970, A050336, A063834, A065026, A066739, A281113, A284639, A318948, A318949.

with(numtheory):
b:= proc(n, k) option remember;
      `if`(n>k, 0, 1) +`if`(isprime(n), 0,
      add(`if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n}))
    end:
a:= proc(n) option remember;
      `if`(n=0, 1, add(add(d*b(d, d), d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60); # Alois P. Heinz, Apr 22 2012

p[ n_, 1 ] := If[ n==1, 1, 0 ]; p[ 1, k_ ] := 1; p[ n_, k_ ] := p[ n, k ]=p[ n, k-1 ]+If[ Mod[ n, k ]==0, p[ n/k, k ], 0 ]; A001055[ n_ ] := p[ n, n ]; a[ n_, 1 ] := 1; a[ 0, k_ ] := 1; a[ n_, k_ ] := If[ k>n, a[ n, n ], a[ n, k ]=a[ n, k-1 ]+Sum[ Binomial[ A001055[ k ]+r-1, r ]a[ n-k*r, k-1 ], {r, 1, Floor[ n/k ]} ] ]; a[ n_ ] := a[ n, n ]; (* p[ n, k ]=number of factorizations of n with factors <= k. a[ n, k ]=number of representations of n as a sum of products of positive integers, with summands <= k *)
b[n_, k_] := b[n, k] = If[n>k, 0, 1] + If[PrimeQ[n], 0, Sum[If[d>k, 0, b[n/d, d]], {d, Divisors[n] ~Complement~ {1, n}}]]; a[0] = 1; a[n_] := a[n] = If[n == 0, 1, Sum[DivisorSum[j, #*b[#, #]&]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Nov 10 2015, after Alois P. Heinz *)
facs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#1,d]&)/@Select[facs[n/d],Min@@#1>=d&],{d,Rest[Divisors[n]]}]];
Table[Length[Union[Sort/@Join@@Table[Tuples[facs/@ptn],{ptn,IntegerPartitions[n]}]]],{n,50}] (* Gus Wiseman, Sep 05 2018 *)

from sympy.core.cache import cacheit
from sympy import divisors, isprime
@cacheit
def b(n, k): return (0 if n>k else 1) + (0 if isprime(n) else sum([0 if d>k else b(n//d, d) for d in divisors(n)[1:-1]]))
@cacheit
def a(n): return 1 if n==0 else sum(sum(d*b(d, d) for d in divisors(j))*a(n - j)  for j in range(1, n + 1))//n
print([a(n) for n in range(61)]) # Indranil Ghosh, Aug 19 2017, after Maple code

a(n) = Sum_{pi} Product_{m=1..n} binomial(k(m)+A001055(m)-1, k(m)), where pi runs through all partitions k(1) + 2 * k( 2) + ... + n * k(n) = n. a(n)=1/n*Sum_{m=1..n} a(n-m)*b(m), n > 0, a(0)=1, b(m)=Sum_{d|m} d*A001055(d). Euler transform of A001055(n): Product_{m=1..infinity} (1-x^m)^(-A001055(m)). - Vladeta Jovovic, Jan 21 2002

Edited by Dean Hickerson, Jan 19 2002

A339564 Number of ways to choose a distinct factor in a factorization of n (pointed factorizations).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 7, 1, 3, 3, 7, 1, 7, 1, 7, 3, 3, 1, 14, 2, 3, 4, 7, 1, 10, 1, 12, 3, 3, 3, 17, 1, 3, 3, 14, 1, 10, 1, 7, 7, 3, 1, 26, 2, 7, 3, 7, 1, 14, 3, 14, 3, 3, 1, 25, 1, 3, 7, 19, 3, 10, 1, 7, 3, 10, 1, 36, 1, 3, 7, 7, 3, 10, 1, 26, 7, 3

Offset: 1

Gus Wiseman, Apr 10 2021

nonn

			The pointed factorizations of n for n = 2, 4, 6, 8, 12, 24, 30:
  ((2))  ((4))    ((6))    ((8))      ((12))     ((24))       ((30))
         ((2)*2)  ((2)*3)  ((2)*4)    ((2)*6)    ((3)*8)      ((5)*6)
                  (2*(3))  (2*(4))    (2*(6))    (3*(8))      (5*(6))
                           ((2)*2*2)  ((3)*4)    ((4)*6)      ((2)*15)
                                      (3*(4))    (4*(6))      (2*(15))
                                      ((2)*2*3)  ((2)*12)     ((3)*10)
                                      (2*2*(3))  (2*(12))     (3*(10))
                                                 ((2)*2*6)    ((2)*3*5)
                                                 (2*2*(6))    (2*(3)*5)
                                                 ((2)*3*4)    (2*3*(5))
                                                 (2*(3)*4)
                                                 (2*3*(4))
                                                 ((2)*2*2*3)
                                                 (2*2*2*(3))

The additive version is A000070 (strict: A015723).
The unpointed version is A001055 (strict: A045778, ordered: A074206, listed: A162247).
Allowing point (1) gives A057567.
Choosing a position instead of value gives A066637.
The ordered additive version is A336875.
A000005 counts divisors.
A001787 count normal multisets with a selected position.
A001792 counts compositions with a selected position.
A006128 counts partitions with a selected position.
A066186 count strongly normal multisets with a selected position.
A254577 counts ordered factorizations with a selected position.
Cf. A007716, A050336, A281113, A281116, A292886, A293627.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Sum[Length[Union[fac]],{fac,facs[n]}],{n,50}]

a(n) = A057567(n) - A001055(n).

a(n) = Sum_{d|n, d>1} A001055(n/d).

A318566 Number of non-isomorphic multiset partitions of multiset partitions of multisets of size n.

Original entry on oeis.org

1, 6, 21, 104, 452, 2335, 11992, 66810, 385101, 2336352, 14738380, 96831730, 659809115, 4657075074, 33974259046, 255781455848, 1984239830571, 15839628564349, 129951186405574, 1094486382191624, 9453318070371926, 83654146992936350, 757769011659766015, 7020652591448497490

Offset: 1

Gus Wiseman, Aug 29 2018

nonn

			Non-isomorphic representatives of the a(3) = 21 multiset partitions of multiset partitions:
  {{{1,1,1}}}
  {{{1,1,2}}}
  {{{1,2,3}}}
  {{{1},{1,1}}}
  {{{1},{1,2}}}
  {{{1},{2,3}}}
  {{{2},{1,1}}}
  {{{1},{1},{1}}}
  {{{1},{1},{2}}}
  {{{1},{2},{3}}}
  {{{1}},{{1,1}}}
  {{{1}},{{1,2}}}
  {{{1}},{{2,3}}}
  {{{2}},{{1,1}}}
  {{{1}},{{1},{1}}}
  {{{1}},{{1},{2}}}
  {{{1}},{{2},{3}}}
  {{{2}},{{1},{1}}}
  {{{1}},{{1}},{{1}}}
  {{{1}},{{1}},{{2}}}
  {{{1}},{{2}},{{3}}}

Cf. A001970, A007716, A050336, A050338, A255906, A269134, A317533, A317791, A318564, A318565.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
dubnorm[m_]:=First[Union[Table[Map[Sort,m/.Rule@@@Table[{Union[Flatten[m]][[i]],Union[Flatten[m]][[perm[[i]]]]},{i,Length[perm]}],{0,2}],{perm,Permutations[Union[Flatten[m]]]}]]];
Table[Length[Union[dubnorm/@Join@@mps/@Join@@mps/@strnorm[n]]],{n,5}]

\\ See links in A339645 for combinatorial species functions.
seq(n)={my(A=sExp(symGroupSeries(n))); NumUnlabeledObjsSeq(sCartProd(A, sExp(A)-1))} \\ Andrew Howroyd, Dec 30 2020

Terms a(8) and beyond from Andrew Howroyd, Dec 30 2020

A340655 Number of twice-balanced factorizations of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
(1) the number of factors;
(2) the maximum image of A001222 over the factors;
(3) A001221(n).

			The twice-balanced factorizations for n = 12, 120, 360, 480, 900, 2520:
  2*6   3*5*8    5*8*9     2*8*30    2*6*75    2*2*7*90
  3*4   2*2*30   2*4*45    3*8*20    2*9*50    2*3*5*84
        2*3*20   2*6*30    4*4*30    3*4*75    2*3*7*60
        2*5*12   2*9*20    4*6*20    3*6*50    2*5*7*36
                 3*4*30    4*8*15    4*5*45    3*3*5*56
                 3*6*20    5*8*12    5*6*30    3*3*7*40
                 3*8*15    6*8*10    5*9*20    3*5*7*24
                 4*5*18    2*12*20   2*10*45   2*2*2*315
                 5*6*12    4*10*12   2*15*30   2*2*3*210
                 2*10*18             2*18*25   2*2*5*126
                 2*12*15             3*10*30   2*3*3*140
                 3*10*12             3*12*25
                                     3*15*20
                                     5*10*18
                                     5*12*15

The co-balanced version is A340596.
The version for unlabeled multiset partitions is A340652.
The balanced version is A340653.
The cross-balanced version is A340654.
Positions of zeros are A340656.
Positions of nonzero terms are A340657.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A045778 counts strict factorizations.
A303975 counts distinct prime factors in prime indices.
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 have an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A003963, A050336, A117409, A320655, A320656, A324518, A339846, A339890, A340607, 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[#]==PrimeNu[n]==Max[PrimeOmega/@#]&]],{n,30}]

cp's OEIS Frontend

A050337 Number of factorizations with one level of parentheses indexed by prime signatures. A050336(A025487).

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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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A001970 Functional determinants; partitions of partitions; Euler transform applied twice to all 1's sequence.

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A281113 Number of twice-factorizations of n. Number of ways to choose a postpositive factorization of each part of a postpositive factorization of n.

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A292504 Number of orderless tree-factorizations of n.

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A066739 Number of representations of n as a sum of products of positive integers. 1 is not allowed as a factor, unless it is the only factor. Representations which differ only in the order of terms or factors are considered equivalent.

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A339564 Number of ways to choose a distinct factor in a factorization of n (pointed factorizations).

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