A096276 -id:A096276 - 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

A057567 Number of partitions of n where the product of parts divides n.

Original entry on oeis.org

1, 2, 2, 4, 2, 5, 2, 7, 4, 5, 2, 11, 2, 5, 5, 12, 2, 11, 2, 11, 5, 5, 2, 21, 4, 5, 7, 11, 2, 15, 2, 19, 5, 5, 5, 26, 2, 5, 5, 21, 2, 15, 2, 11, 11, 5, 2, 38, 4, 11, 5, 11, 2, 21, 5, 21, 5, 5, 2, 36, 2, 5, 11, 30, 5, 15, 2, 11, 5, 15, 2, 52, 2, 5, 11, 11, 5, 15, 2, 38, 12, 5, 2, 36, 5, 5, 5, 21

Offset: 1

Leroy Quet, Oct 04 2000

nonn

a(n) depends only on prime signature of n (cf. A025487). So a(24) = a(375) since 24=2^3*3 and 375=3*5^3 both have prime signature (3,1). - Christian G. Bower, Jun 03 2005

			From _Gus Wiseman_, Jul 04 2019: (Start)
The a(1) = 1 through a(9) = 5 partitions are the following. The Heinz numbers of these partitions are given by A326155.
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (11111)  (321)     (1111111)  (4211)
                    (211)            (3111)               (22211)
                    (1111)           (21111)              (41111)
                                     (111111)             (221111)
                                                          (2111111)
                                                          (11111111)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Cf. A000070, A000110, A001055 (Mobius transform), A002110, A025487, A057568, A108464, A113309, A131802.
Any prime numbered column of array A108461.
Cf. A028422, A096276, A114324, A318950, A319000, A319005, A326152, A326155.

Table[Function[m, Count[Map[Times @@ # &, IntegerPartitions[m]], P_ /; Divisible[m, P]] - Boole[n == 1]]@ Apply[Times, #] &@ MapIndexed[Prime[First@ #2]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]], {n, 88}] (* Michael De Vlieger, Aug 16 2017 *)

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) \\ This function from Michael B. Porter, Oct 29 2009
A057567(n) = sumdiv(n, d, A001055(d)); \\ After Jovovic's formula. Antti Karttunen, May 25 2017

from sympy import divisors, isprime
def T(n, m):
    if isprime(n): return 1 if n <= m else 0
    A = (d for d in divisors(n) if 1 < d < n and d <= m)
    s = sum(T(n // d, d) for d in A)
    return s + 1 if n <= m else s
def a001055(n): return T(n, n)
def a(n): return sum(a001055(d) for d in divisors(n))
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Aug 19 2017

a(n) = Sum_{d|n} A001055(d). - Vladeta Jovovic, Nov 19 2000
a(A025487(n)) = A108464(n).
a(p^k) = A000070(k).
a(A002110(n)) = A000110(n+1).
Dirichlet g.f.: zeta(s) * Product_{k>=2} 1/(1 - 1/k^s). - Ilya Gutkovskiy, Nov 03 2020

More terms from James Sellers, Oct 09 2000

More terms from Vladeta Jovovic, Nov 19 2000

A057568 Number of partitions of n where n divides the product of the parts.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 6, 5, 5, 1, 22, 1, 11, 23, 80, 1, 113, 1, 150, 85, 45, 1, 737, 226, 84, 809, 726, 1, 1787, 1, 4261, 735, 260, 1925, 9567, 1, 437, 1877, 16402, 1, 14630, 1, 9861, 33057, 1152, 1, 102082, 19393, 57330, 10159, 30706, 1, 207706, 47927, 200652

Offset: 1

Leroy Quet, Oct 04 2000

nonn

			From _Gus Wiseman_, Jul 04 2019: (Start)
The a(1) = 1 through a(9) = 5 partitions are the following. The Heinz numbers of these partitions are given by A326149.
  (1)  (2)  (3)  (4)   (5)  (6)    (7)  (8)      (9)
                 (22)       (321)       (44)     (63)
                                        (422)    (333)
                                        (2222)   (3321)
                                        (4211)   (33111)
                                        (22211)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (terms n=1..73 from Antti Karttunen)

Cf. A028422, A057567, A096276, A113309, A114324, A318950, A319000, A319005, A326149, A326152.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=1, 1, 0), `if`(i<1, 0, b(n, i-1, t)+
      `if`(i>n, 0, b(n-i, min(i, n-i), t/igcd(i, t)))))
    end:
a:= n-> `if`(isprime(n), 1, b(n$3)):
seq(a(n), n=1..70);  # Alois P. Heinz, Dec 20 2017

Table[Length[Select[IntegerPartitions[n],Divisible[Times@@#,n]&]],{n,20}] (* Gus Wiseman, Jul 04 2019 *)
b[n_, i_, t_] := b[n, i, t] = If[n == 0, If[t == 1, 1, 0], If[i < 1, 0, b[n, i - 1, t] + If[i > n, 0, b[n - i, Min[i, n - i], t/GCD[i, t]]]]];
a[n_] := If[PrimeQ[n], 1, b[n, n, n]];
Array[a, 70] (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

;; This is a naive algorithm that scans over all partitions of each n. For fold_over_partitions_of see A000793.
(define (A057568 n) (let ((z (list 0))) (fold_over_partitions_of n 1 * (lambda (partprod) (if (zero? (modulo partprod n)) (set-car! z (+ 1 (car z)))))) (car z)))
;; Antti Karttunen, Dec 20 2017

More terms from James Sellers, Oct 09 2000

A325044 Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 22, 23, 24, 26, 28, 29, 30, 31, 32, 34, 36, 37, 38, 40, 41, 43, 44, 46, 47, 48, 52, 53, 56, 58, 59, 60, 61, 62, 64, 67, 68, 71, 72, 73, 74, 76, 79, 80, 82, 83, 84, 86, 88, 89, 92, 94, 96, 97, 101

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is less than or equal to their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A096276.

			The sequence of terms together with their prime indices begins:
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  16: {1,1,1,1}
  17: {7}
  18: {1,2,2}
  19: {8}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  24: {1,1,1,2}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325038, A325041, A325042.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Times@@primeMS[#]<=Plus@@primeMS[#]&]

A003963(a(n)) <= A056239(a(n)).
a(n) = A325038(n)/2.
Union of A301987 and A325038.

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 *)

A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product.

Original entry on oeis.org

4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 34, 36, 38, 40, 44, 46, 48, 52, 56, 58, 60, 62, 64, 68, 72, 74, 76, 80, 82, 86, 88, 92, 94, 96, 104, 106, 112, 116, 118, 120, 122, 124, 128, 134, 136, 142, 144, 146, 148, 152, 158, 160, 164, 166, 168, 172

Offset: 1

Gus Wiseman, Mar 25 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1) * ... * prime(y_k), so these are numbers whose product of prime indices (A003963) is less than their sum of prime indices (A056239).

The enumeration of these partitions by sum is given by A096276 shifted once to the right.

			The sequence of terms together with their prime indices begins:
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  32: {1,1,1,1,1}
  34: {1,7}
  36: {1,1,2,2}
  38: {1,8}
  40: {1,1,1,3}
  44: {1,1,5}
  46: {1,9}
  48: {1,1,1,1,2}

Cf. A000720, A003963, A056239, A112798, A178503, A175508, A301987, A319000.

Cf. A325032, A325033, A325036, A325037, A325041, A325042, A325044.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Times@@primeMS[#]

A003963(a(n)) < A056239(a(n)).

a(n) = 2 * A325044(n).

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}]

A379721 Numbers whose prime indices have sum <= product.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 30, 31, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 100, 101, 102, 103

Offset: 1

Gus Wiseman, Jan 05 2025

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
Partitions of this type are counted by A319005.
The complement is A325038.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}

The case of equality is A301987, inequality A325037.
Nonpositive positions in A325036.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A379681 gives sum plus product of prime indices, firsts A379682.
Counting and ranking multisets by comparing sum and product:
- same: A001055 (strict A045778), ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A326156, A326172, A379733
- greater: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721 (this)
- different: A379736, ranks A379722, see A111133
Cf. A000720, A003963, A046022, A075254, A075255, A178503, A175508, A319000, A325034, A325035, A379720.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Total[prix[#]]<=Times@@prix[#]&]

Number k such that A056239(k) <= A003963(k).

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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A057567 Number of partitions of n where the product of parts divides n.

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A057568 Number of partitions of n where n divides the product of the parts.

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A325044 Heinz numbers of integer partitions whose sum of parts is greater than or equal to their product.

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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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A325038 Heinz numbers of integer partitions whose sum of parts is greater than their product.

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