A326155 -id:A326155 - cp's OEIS frontend

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

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

A326149 Numbers whose product of prime indices is divisible by their sum of prime indices.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 43, 47, 49, 53, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 108, 109, 113, 125, 127, 131, 137, 139, 149, 150, 151, 154, 157, 163, 165, 167, 169, 173, 179, 181, 190, 191, 193, 197

Offset: 1

Gus Wiseman, Jun 09 2019

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.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of integer partitions whose product of parts is divisible by their sum of parts. The enumeration of these partitions by sum is given by A057568.

			The sequence of terms together with their prime indices begins:
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   17: {7}
   19: {8}
   23: {9}
   29: {10}
   30: {1,2,3}
   31: {11}
   37: {12}
   41: {13}
   43: {14}
   47: {15}
   49: {4,4}
   53: {16}
   59: {17}

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

Satisfies A056239(a(n))|A003963(a(n)).
The nonprime case is A326150, with squarefree case A326158.
Cf. A000720, A001222, A057567, A057568, A112798, A301987.
Cf. A325037, A325042, A325044, A326151, A326153/A326154, A326155, A326156.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,100],Divisible[Times@@primeMS[#],Plus@@primeMS[#]]&]

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).

A379722 Numbers whose prime indices do not have the same sum as product.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 90, 91, 92, 93

Offset: 1

Gus Wiseman, Jan 08 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 A379736.
The complement is A301987, counted by A001055.

			The terms together with their prime indices begin:
    1: {}
    4: {1,1}
    6: {1,2}
    8: {1,1,1}
   10: {1,3}
   12: {1,1,2}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   18: {1,2,2}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   24: {1,1,1,2}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}

Nonzeros of 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.
A324851 finds numbers > 1 divisible by the sum of their prime indices.
A379666 counts partitions by sum and product, without 1's A379668.
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 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
- different: A379736, ranks A379722 (this)
Cf. A000720, A003963, A025147, A075254, A111133, A178503, A319000, A325041.

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

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}]

A326156 Number of nonempty subsets of {1..n} whose product is divisible by their sum.

Original entry on oeis.org

0, 1, 2, 4, 5, 10, 19, 34, 64, 129, 267, 541, 1104, 2253, 4694, 9804, 18894, 38539, 76063, 155241, 311938, 636120, 1299869, 2653853, 5183363, 10272289, 20958448, 40945577, 81745769, 167048919, 329598054, 671038751, 1301431524, 2618590422, 5305742557, 10582105199, 20660489585, 42075929255, 85443680451, 172057673225, 338513788818

Offset: 0

Gus Wiseman, Jun 11 2019

nonn

			The a(1) = 1 through a(6) = 19 subsets:
  {1}  {1}  {1}      {1}      {1}          {1}
       {2}  {2}      {2}      {2}          {2}
            {3}      {3}      {3}          {3}
            {1,2,3}  {4}      {4}          {4}
                     {1,2,3}  {5}          {5}
                              {1,2,3}      {6}
                              {1,4,5}      {3,6}
                              {2,3,5}      {1,2,3}
                              {3,4,5}      {1,4,5}
                              {1,2,3,4,5}  {2,3,5}
                                           {2,4,6}
                                           {3,4,5}
                                           {4,5,6}
                                           {1,2,3,6}
                                           {1,3,5,6}
                                           {3,4,5,6}
                                           {1,2,3,4,5}
                                           {1,2,3,4,6}
                                           {2,3,4,5,6}

Bert Dobbelaere, C++ program

Cf. A053632, A057567, A057568, A059529, A063865, A301987, A326150, A326151, A326153/A326154, A326155, A326158, A326178, A326179, A326180.

Table[Length[Select[Subsets[Range[n],{1,n}],Divisible[Times@@#,Plus@@#]&]],{n,0,10}]

a(21)-a(30) from Alois P. Heinz, Jun 13 2019

a(31)-a(40) from Bert Dobbelaere, Jun 22 2019

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.

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

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

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A326149 Numbers whose product of prime indices is divisible by their sum of prime indices.

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A379721 Numbers whose prime indices have sum <= product.

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A379722 Numbers whose prime indices do not have the same sum as product.

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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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A326156 Number of nonempty subsets of {1..n} whose product is divisible by their sum.

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

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

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