A319916 -id:A319916 - 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

A319000 Regular triangle where T(n,k) is the number of finite multisets of positive integers with product n and sum k.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 22 2018

			Triangle begins:
  1
  0 1
  0 0 1
  0 0 0 2
  0 0 0 0 1
  0 0 0 0 1 2
  0 0 0 0 0 0 1
  0 0 0 0 0 2 2 3
  0 0 0 0 0 1 1 1 2
  0 0 0 0 0 0 1 1 1 2
  0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 2 3 3 3 3 4
  0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 1 1 1 1 1 2
  0 0 0 0 0 0 0 1 1 1 1 1 1 1 2
  0 0 0 0 0 0 0 3 3 4 4 4 4 4 4 5
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 1 2 2 3 3 3 3 3 3 3 4
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 2 2 2 3 3 3 3 3 3 3 3 4
Row 12 {0,0,0,0,0,0,2,3,3,3,3,4} corresponds to the partitions (C = 12):
. . . . . . (43)  (62)   (621)   (6211)   (62111)    (C)
            (322) (431)  (4311)  (43111)  (431111)   (621111)
                  (3221) (32211) (322111) (3221111)  (4311111)
                                          (32211111)

Row sums are A319916. Column sums are A319005. Last column is A001055.

Cf. A000041, A002865, A069016, A096276, A301987, A318950, A319057.

Table[Length[Select[IntegerPartitions[k],Times@@#==n&]],{n,20},{k,n}]

A319005 Number of integer partitions of n whose product of parts is >= n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 5, 7, 13, 18, 28, 40, 60, 80, 113, 152, 205, 266, 353, 454, 590, 751, 959, 1210, 1529, 1905, 2381, 2953, 3658, 4501, 5539, 6772, 8278, 10065, 12230, 14801, 17893, 21544, 25921, 31089, 37240, 44478, 53068, 63150, 75063, 89018, 105438, 124632

Offset: 0

Gus Wiseman, Oct 22 2018

nonn

			The a(1) = 1 through a(9) = 18 partitions:
  (1)  (2)  (3)  (4)   (5)   (6)    (7)     (8)      (9)
                 (22)  (32)  (33)   (43)    (44)     (54)
                             (42)   (52)    (53)     (63)
                             (222)  (322)   (62)     (72)
                             (321)  (331)   (332)    (333)
                                    (421)   (422)    (432)
                                    (2221)  (431)    (441)
                                            (521)    (522)
                                            (2222)   (531)
                                            (3221)   (621)
                                            (3311)   (3222)
                                            (4211)   (3321)
                                            (22211)  (4221)
                                                     (4311)
                                                     (5211)
                                                     (22221)
                                                     (32211)
                                                     (33111)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.

Column sums of A319000.

Cf. A001055, A002865, A069016, A096276, A301987, A318950, A319057, A319916.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1, `if`(p>1,
      0, 1), b(n, i-1, p) +b(n-i, min(i, n-i), max(p/i, 1)))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 22 2018

Table[Length[Select[IntegerPartitions[n],Times@@#>=n&]],{n,50}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, If[p > 1, 0, 1],
     b[n, i - 1, p] + b[n - i, Min[i, n - i], Max[p/i, 1]]];
a[n_] := b[n, n, n];
a /@ Range[0, 50] (* Jean-François Alcover, May 11 2021, after Alois P. Heinz *)

A096276 Number of partitions of n with a product <=n.

Original entry on oeis.org

0, 1, 2, 3, 5, 6, 8, 9, 12, 14, 16, 17, 21, 22, 24, 26, 31, 32, 36, 37, 41, 43, 45, 46, 53, 55, 57, 60, 64, 65, 70, 71, 78, 80, 82, 84, 93, 94, 96, 98, 105, 106, 111, 112, 116, 120, 122, 123, 135, 137, 141, 143, 147, 148, 155, 157, 164, 166, 168, 169, 180, 181, 183, 187

Offset: 0

Jon Perry, Jun 23 2004

nonn

The Heinz numbers of these partitions are given by A325044. - Gus Wiseman, Mar 27 2019

			a(6)=8 as we can have 6, 51, 411, 321, 3111, 2211, 21111, 111111, rejecting 42, 33 and 222.
From _Gus Wiseman_, Mar 27 2019: (Start)
The a(1) = 1 through a(8) = 12 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (21)   (22)    (41)     (51)      (61)       (71)
             (111)  (31)    (221)    (321)     (511)      (611)
                    (211)   (311)    (411)     (3211)     (4211)
                    (1111)  (2111)   (2211)    (4111)     (5111)
                            (11111)  (3111)    (22111)    (22211)
                                     (21111)   (31111)    (32111)
                                     (111111)  (211111)   (41111)
                                               (1111111)  (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)
(End)

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).

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.
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.
A. Oppenheim, On an Arithmetic Function, Journal of the London Mathematical Society, 1926, 10, Vol. s1-1, Iss. 4, 205-211.
A. Oppenheim, On an Arithmetic Function (II), Journal of the London Mathematical Society, 1927, 04, Vol. s1-2, Iss. 2, 123-130.
Csaba Sándor and Maciej Zakarczemny, Equal Sum and Product Problem III, arXiv:2405.11600 [math.NT], 2024.

Cf. A001055, A028422, A114324, A301987, A319000, A319005, A319916, A325044.

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:= proc(n) option remember; `if`(n=0, 0, a(n-1)+g(n$2)) end:
seq(a(n), n=0..100);  # Alois P. Heinz, Feb 26 2023

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]; Join[{0}, Accumulate[Array[a, 100]]] (* using program from A001055, T. D. Noe, Apr 11 2011 *)
Table[Length[Select[IntegerPartitions[n],Times@@#<=n&]],{n,0,20}] (* Gus Wiseman, Mar 27 2019 *)

{ bla(n,m,v,z)=v=concat(v,m); if(!n,x=prod(k=1,length(v),v[k]); if (x<=z,c++), for(i=1,min(m,n),bla(n-i,i,v,z))); }
q(n)=c=0;for(i=1,n,bla(n-i,i,[],n));print1(c, ", ");
for(i=0,40,q(i))

For n>1, a(n) = a(n-1)+1 iff n is prime.
Partial sums of A001055. - Vladeta Jovovic, Jun 24 2004
a(n) ~ n * exp(2*sqrt(log(n))) / (2*sqrt(Pi) * (log(n))^(3/4)) [Oppenheim, 1927]. - Vaclav Kotesovec, May 23 2020

More terms from Vladeta Jovovic, Jun 24 2004

A114324 Number of partitions of n with a product greater than n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 3, 6, 10, 16, 26, 39, 56, 79, 111, 150, 200, 265, 349, 453, 586, 749, 957, 1209, 1522, 1903, 2379, 2950, 3654, 4500, 5534, 6771, 8271, 10063, 12228, 14799, 17884, 21543, 25919, 31087, 37233, 44477, 53063, 63149, 75059, 89014, 105436, 124631

Offset: 0

Giovanni Resta, Feb 06 2006

nonn

The Heinz numbers of these partitions are given by A325037. - Gus Wiseman, Mar 27 2019

			a(6) = 3 since there are 3 partitions of 6 with product greater than 6: {3,3}, {2,2,2}, {4,2}.
From _Gus Wiseman_, Mar 27 2019: (Start)
The a(5) = 1 through a(9) = 16 partitions:
  (32)  (33)   (43)    (44)    (54)
        (42)   (52)    (53)    (63)
        (222)  (322)   (62)    (72)
               (331)   (332)   (333)
               (421)   (422)   (432)
               (2221)  (431)   (441)
                       (521)   (522)
                       (2222)  (531)
                       (3221)  (621)
                       (3311)  (3222)
                               (3321)
                               (4221)
                               (4311)
                               (5211)
                               (22221)
                               (32211)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
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.

Cf. A001055, A028422, A096276, A114324, A301987, A319000, A319005, A319916, A325037, A325038, A325044.

<< DiscreteMath`Combinatorica`; lst=Table[Length@Select[Partitions[n], (Times @@ # > n) &],{n,50}]
Table[Length[Select[IntegerPartitions[n],Times@@#>n&]],{n,0,20}] (* Gus Wiseman, Mar 27 2019 *)

a(0) = 1 prepended by Gus Wiseman, Mar 27 2019

A318950 Regular triangle where T(n,k) is the number of factorizations of n into factors > 1 with sum k.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Oct 22 2018

			Triangle begins:
  0
  0 1
  0 0 1
  0 0 0 2
  0 0 0 0 1
  0 0 0 0 1 1
  0 0 0 0 0 0 1
  0 0 0 0 0 2 0 1
  0 0 0 0 0 1 0 0 1
  0 0 0 0 0 0 1 0 0 1
  0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 2 1 0 0 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 1 0 0 0 0 1
  0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 3 0 1 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
  0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0 0 1
Row 12 {0,0,0,0,0,0,2,1,0,0,0,1} corresponds to the factorizations:
  . . . . . . (3*4)   (2*6) . . . (12)
              (2*2*3)

Row sums are A001055. Column sums are A002865.

Cf. A069016, A096276, A301987, A319000, A319005, A319057, A319916.

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],Total[#]==k&]],{n,20},{k,n}]

A379733 Number of strict integer partitions of n whose product of parts is a multiple of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 5, 1, 5, 7, 7, 1, 12, 1, 20, 15, 11, 1, 48, 12, 16, 33, 61, 1, 121, 1, 105, 67, 34, 126, 292, 1, 49, 128, 471, 1, 522, 1, 387, 751, 96, 1, 1556, 246, 792, 422, 869, 1, 2126, 1191, 2904, 726, 240, 1, 6393, 1, 321, 5460, 6711

Offset: 1

Gus Wiseman, Jan 07 2025

nonn

Partitions of this type are ranked by the squarefree terms of A326149.

			The a(n) partitions for n = 1, 6, 10, 12, 15, 18:
  (1)  (6)      (10)     (12)       (15)         (18)
       (3,2,1)  (5,3,2)  (5,4,3)    (6,5,4)      (12,6)
                (5,4,1)  (6,4,2)    (7,5,3)      (9,5,4)
                         (8,3,1)    (9,5,1)      (9,6,3)
                         (6,3,2,1)  (10,3,2)     (9,7,2)
                                    (6,5,3,1)    (9,8,1)
                                    (5,4,3,2,1)  (6,5,4,3)
                                                 (7,6,3,2)
                                                 (8,6,3,1)
                                                 (9,4,3,2)
                                                 (9,6,2,1)
                                                 (12,3,2,1)

The non-strict opposite version is A057567, ranks A326155.
The non-strict version is A057568, ranks A326149.
The case of partitions without 1's is A379735, non-strict A379734.
A319005 counts partitions with product >= sum, ranks A379721.
A114324 counts partitions with product greater than sum, ranks A325037.
Cf. A001055, A003963, A069016, A096276, A111133, A318950, A319000, A319057, A319916, A326152, A379720.

b:= proc(n, i, t) option remember; `if`(i*(i+1)/2 `if`(isprime(n), 1, b(n$3)):
seq(a(n), n=1..70);  # Alois P. Heinz, Jan 07 2025

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&Divisible[Times@@#,n]&]],{n,30}]

A379736 Number of integer partitions of n whose product of parts is not n.

Original entry on oeis.org

1, 0, 1, 2, 3, 6, 9, 14, 19, 28, 40, 55, 73, 100, 133, 174, 226, 296, 381, 489, 623, 790, 1000, 1254, 1568, 1956, 2434, 3007, 3714, 4564, 5599, 6841, 8342, 10141, 12308, 14881, 17968, 21636, 26013, 31183, 37331, 44582, 53169, 63260, 75171, 89130, 105556

Offset: 0

Gus Wiseman, Jan 07 2025

nonn

These partitions are ranked by A379722, complement A301987.

			The a(2) = 1 through a(7) = 14 partitions:
  (11)  (21)   (31)    (32)     (33)      (43)
        (111)  (211)   (41)     (42)      (52)
               (1111)  (221)    (51)      (61)
                       (311)    (222)     (322)
                       (2111)   (411)     (331)
                       (11111)  (2211)    (421)
                                (3111)    (511)
                                (21111)   (2221)
                                (111111)  (3211)
                                          (4111)
                                          (22111)
                                          (31111)
                                          (211111)
                                          (1111111)

The complement is counted by A001055.
The strict case is A111133 (except first term).
A000041 counts integer partitions, strict A000009.
A002865 counts partitions into parts > 1, see A379734, strict A379735.
A324851 finds numbers > 1 divisible by the sum of their prime indices.
A379666 counts partitions by sum and product, without 1's A379668.
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, A379720
- less or equal: A319005, ranks A379721, see A025147
- different: A379736 (this), ranks A379722
Cf. A028422, A069016, A319000, A319916, A325036, A325041, A326152, A379671, A379678.

Table[Length[Select[IntegerPartitions[n],Times@@#!=n&]],{n,0,30}]

a(n) = A000041(n) - A001055(n).

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 *)

A379720 Except a(0)=1 and a(4)=0, number of integer partitions of n with no 1's and at least two parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 3, 3, 6, 7, 11, 13, 20, 23, 33, 40, 54, 65, 87, 104, 136, 164, 209, 252, 319, 382, 477, 573, 707, 846, 1038, 1237, 1506, 1793, 2166, 2572, 3093, 3659, 4377, 5169, 6152, 7244, 8590, 10086, 11913, 13958, 16423, 19195, 22518, 26251, 30700, 35716

Offset: 0

Gus Wiseman, Jan 06 2025

nonn

Also partitions of n such that all parts are > 1 and have product > n.

Allowing 1's gives A114324, ranks A325037. The strict case is A318029 (except first term).

			The a(5) = 1 through a(11) = 13 partitions:
  (3,2)  (3,3)    (4,3)    (4,4)      (5,4)      (5,5)        (6,5)
         (4,2)    (5,2)    (5,3)      (6,3)      (6,4)        (7,4)
         (2,2,2)  (3,2,2)  (6,2)      (7,2)      (7,3)        (8,3)
                           (3,3,2)    (3,3,3)    (8,2)        (9,2)
                           (4,2,2)    (4,3,2)    (4,3,3)      (4,4,3)
                           (2,2,2,2)  (5,2,2)    (4,4,2)      (5,3,3)
                                      (3,2,2,2)  (5,3,2)      (5,4,2)
                                                 (6,2,2)      (6,3,2)
                                                 (3,3,2,2)    (7,2,2)
                                                 (4,2,2,2)    (3,3,3,2)
                                                 (2,2,2,2,2)  (4,3,2,2)
                                                              (5,2,2,2)
                                                              (3,2,2,2,2)

For <= instead of < we have A002865 = partitions into parts > 1.
These partitions have ranks A071904 (except initial terms).
Set a(4) = 1 to get A083751.
A000041 counts integer partitions, strict A000009.
A379668 counts partitions without 1's by sum and product.
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, A318950, A319000, A319916, A325036, A325041, A326152, A326178, A379666, A379678.

Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&Plus@@#

Except for n = 0 and n = 4, we have a(n) = A002865(n) - 1.

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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A319000 Regular triangle where T(n,k) is the number of finite multisets of positive integers with product n and sum k.

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A319005 Number of integer partitions of n whose product of parts is >= n.

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A096276 Number of partitions of n with a product <=n.

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A114324 Number of partitions of n with a product greater than n.

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A318950 Regular triangle where T(n,k) is the number of factorizations of n into factors > 1 with sum k.

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A379733 Number of strict integer partitions of n whose product of parts is a multiple of n.

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