A325828 -id:A325828 - cp's OEIS frontend

A325792 Positive integers with as many proper divisors as the sum of their prime indices.

1, 2, 4, 6, 8, 16, 18, 20, 32, 42, 54, 56, 64, 100, 128, 162, 176, 204, 234, 256, 260, 294, 308, 315, 350, 392, 416, 486, 500, 512, 690, 696, 798, 920, 1024, 1026, 1064, 1088, 1116, 1122, 1190, 1365, 1430, 1458, 1496, 1755, 1936, 1968, 2025, 2048, 2058, 2079

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from A325780 in having 204.

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, with sum A056239(n).

			The term 42 is in the sequence because it has 7 proper divisors (1, 2, 3, 6, 7, 14, 21) and its sum of prime indices is also 1 + 2 + 4 = 7.
The sequence of terms together with their prime indices begins:
     1: {}
     2: {1}
     4: {1,1}
     6: {1,2}
     8: {1,1,1}
    16: {1,1,1,1}
    18: {1,2,2}
    20: {1,1,3}
    32: {1,1,1,1,1}
    42: {1,2,4}
    54: {1,2,2,2}
    56: {1,1,1,4}
    64: {1,1,1,1,1,1}
   100: {1,1,3,3}
   128: {1,1,1,1,1,1,1}
   162: {1,2,2,2,2}
   176: {1,1,1,1,5}
   204: {1,1,2,7}
   234: {1,2,2,6}
   256: {1,1,1,1,1,1,1,1}

Positions of 1's in A325794.
Heinz numbers of the partitions counted by A325828.
Cf. A000005, A002033, A056239, A112798, A299702, A304793.
Cf. A325694, A325780, A325781, A325793, A325795, A325796, A325797, A325798.

Select[Range[100],DivisorSigma[0,#]-1==Total[Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]]&]

A325833 Number of integer partitions of n whose number of submultisets is less than n.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 5, 7, 9, 14, 20, 21, 27, 43, 50, 56, 69, 98, 118, 143, 165, 200, 229, 249, 282, 454, 507, 555, 637, 706, 789, 889, 986, 1406, 1567, 1690, 1875, 2396, 2602, 2841, 3078, 3672, 3977, 4344, 4660, 5079, 5488, 5840, 6296, 10424, 11306

Offset: 0

Gus Wiseman, May 29 2019

nonn

The number of submultisets of a partition is the product of its multiplicities, each plus one.

The Heinz numbers of these partitions are given by A325797.

			The a(3) = 1 through a(9) = 14 partitions:
  (3)  (4)   (5)   (6)    (7)    (8)     (9)
       (22)  (32)  (33)   (43)   (44)    (54)
             (41)  (42)   (52)   (53)    (63)
                   (51)   (61)   (62)    (72)
                   (222)  (322)  (71)    (81)
                          (331)  (332)   (333)
                          (511)  (422)   (432)
                                 (611)   (441)
                                 (2222)  (522)
                                         (531)
                                         (621)
                                         (711)
                                         (3222)
                                         (6111)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A002033, A088880, A088881, A098859, A108917, A307699, A325694, A325792, A325797, A325828, A325830, A325831, A325832, A325834, A325836.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
      (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
    end:
a:= n-> add(b(n$2, k), k=0..n-1):
seq(a(n), n=0..55);  # Alois P. Heinz, Aug 17 2019

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])Jean-François Alcover, May 12 2021, after Alois P. Heinz *)

a(n) = A000041(n) - A325832(n).

For n even, a(n) = A325834(n) - A325830(n/2); for n odd, a(n) = A325834(n).

A325830 Number of integer partitions of 2n having exactly 2n submultisets.

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 10, 1, 21, 12, 15, 1, 121, 1, 20, 37, 309, 1, 319, 1, 309, 47, 33, 1, 3435, 30, 38, 405, 593, 1, 1574, 1, 11511, 80, 51, 77, 17552, 1, 56, 92, 13921, 1, 3060, 1, 1439, 2911, 69, 1, 234969, 56, 2044, 126, 1998, 1, 46488, 114, 36615, 137, 87, 1, 141906

Offset: 0

Gus Wiseman, May 25 2019

nonn

If n is odd, there are no integer partitions of n with exactly n submultisets, so this sequence gives only the even-indexed terms.
The number of submultisets of an integer partition is the product of its multiplicities, each plus one.
The Heinz numbers of these partitions are given by A325793.

			The 12 submultisets of the partition (7221) are (), (1), (2), (7), (21), (22), (71), (72), (221), (721), (722), (7221), so (7221) is counted under a(6).
The a(1) = 1 through a(8) = 21 partitions (A = 10, B = 11):
  (2)  (31)  (411)  (431)   (61111)  (4332)    (8111111)  (6532)
                    (521)            (4431)               (6541)
                    (5111)           (5322)               (7432)
                                     (5331)               (7531)
                                     (6411)               (7621)
                                     (7221)               (8431)
                                     (7311)               (8521)
                                     (8211)               (9421)
                                     (33222)              (A321)
                                     (711111)             (44431)
                                                          (53332)
                                                          (63331)
                                                          (64222)
                                                          (73222)
                                                          (76111)
                                                          (85111)
                                                          (92221)
                                                          (94111)
                                                          (A3111)
                                                          (B2111)
                                                          (91111111)

Alois P. Heinz, Table of n, a(n) for n = 0..700 (first 101 terms from Andrew Howroyd)

Cf. A002033, A098859, A108917, A126796, A237999, A325694, A325792, A325793, A325828, A325831, A325832, A325833, A325834, A325836.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
      (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
    end:
a:= n-> `if`(isprime(n), 1, b(2*n$3)):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 16 2019

Table[Length[Select[IntegerPartitions[2*n],Times@@(1+Length/@Split[#])==2*n&]],{n,0,30}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
     If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, r = p/(j + 1);
     Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := If[PrimeQ[n], 1, b[2n, 2n, 2n]];
a /@ Range[0, 60] (* Jean-François Alcover, May 12 2021, after Alois P. Heinz *)

a(n)={if(n<1, 0, my(v=vector(2*n+1, k, vector(2*n))); v[1][1]=1; for(k=1, 2*n, forstep(j=#v, k, -1, for(m=1, (j-1)\k, for(i=1, 2*n\(m+1), v[j][i*(m+1)] += v[j-m*k][i])))); v[#v][2*n])} \\ Andrew Howroyd, Aug 16 2019

a(p) = 1 for prime p. - Andrew Howroyd, Aug 16 2019

Terms a(31) and beyond from Andrew Howroyd, Aug 16 2019

A325836 Number of integer partitions of n having n - 1 different submultisets.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 0, 3, 0, 5, 2, 2, 0, 15, 0, 2, 3, 25, 0, 17, 0, 18, 3, 2, 0, 150, 0, 2, 13, 24, 0, 43, 0, 351, 3, 2, 2, 383, 0, 2, 3, 341, 0, 60, 0, 37, 51, 2, 0, 3733, 0, 31, 3, 42, 0, 460, 1, 633, 3, 2, 0, 1780, 0, 2, 68, 12460, 0, 87, 0, 55, 3

Offset: 0

Gus Wiseman, May 29 2019

nonn

The number of submultisets of a partition is the product of its multiplicities, each plus one.

The Heinz numbers of these partitions are given by A325694.

			The a(3) = 1 through a(13) = 15 partitions (empty columns not shown):
  (3)  (22)  (32)  (322)  (432)   (3322)  (32222)  (4432)
             (41)  (331)  (531)   (4411)  (71111)  (5332)
                   (511)  (621)                    (5422)
                          (3222)                   (5521)
                          (6111)                   (6322)
                                                   (6331)
                                                   (6511)
                                                   (7411)
                                                   (8221)
                                                   (8311)
                                                   (9211)
                                                   (33322)
                                                   (55111)
                                                   (322222)
                                                   (811111)

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

Positions of zeros are A307699.

Cf. A002033, A088880, A088881, A108917, A325694, A325768, A325792, A325798, A325828, A325830, A325833, A325835.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
      (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
    end:
a:= n-> b(n$2,n-1):
seq(a(n), n=0..80);  # Alois P. Heinz, Aug 17 2019

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])==n-1&]],{n,0,30}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
     If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, r = p/(j + 1);
     Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := b[n, n, n-1];
a /@ Range[0, 80] (* Jean-François Alcover, May 12 2021, after Alois P. Heinz *)

A325831 Number of integer partitions of n whose number of submultisets is greater than n.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 5, 8, 10, 16, 21, 35, 40, 58, 84, 120, 141, 199, 255, 347, 447, 592, 772, 1006, 1172, 1504, 1928, 2455, 3061, 3859, 4778, 5953, 7054, 8737, 10742, 13193, 15783, 19241, 23412, 28344, 33951, 40911, 49150, 58917, 70482, 84055, 100069, 118914

Offset: 0

Gus Wiseman, May 25 2019

nonn

The number of submultisets of a partition is the product of its multiplicities, each plus one.

The Heinz numbers of these partitions are given by A325795.

			The a(1) = 1 through a(8) = 10 partitions:
  (1)  (11)  (21)   (211)   (221)    (321)     (421)      (3221)
             (111)  (1111)  (311)    (2211)    (2221)     (3311)
                            (2111)   (3111)    (3211)     (4211)
                            (11111)  (21111)   (4111)     (22211)
                                     (111111)  (22111)    (32111)
                                               (31111)    (41111)
                                               (211111)   (221111)
                                               (1111111)  (311111)
                                                          (2111111)
                                                          (11111111)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A002033, A098859, A126796, A325694, A325792, A325795, A325828, A325830, A325832, A325833, A325834, A325836.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
      (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
    end:
a:= n-> combinat[numbpart](n)-add(b(n$2, k), k=0..n):
seq(a(n), n=0..55);  # Alois P. Heinz, Aug 17 2019

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])>n&]],{n,0,30}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
     If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0,
     Function[w, b[w, Min[w, i-1], p/(j+1)]][n-i*j], 0], {j, 0, n/i}]];
a[n_] := PartitionsP[n] - Sum[b[n, n, k], {k, 0, n}];
a /@ Range[0, 55] (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)

a(n) = A000041(n) - A325834(n).

For n even, a(n) = A325832(n) - A325830(n/2); for n odd, a(n) = A325832(n).

A325834 Number of integer partitions of n whose number of submultisets is less than or equal to n.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 6, 7, 12, 14, 21, 21, 37, 43, 51, 56, 90, 98, 130, 143, 180, 200, 230, 249, 403, 454, 508, 555, 657, 706, 826, 889, 1295, 1406, 1568, 1690, 2194, 2396, 2603, 2841, 3387, 3672, 4024, 4344, 4693, 5079, 5489, 5840, 9731, 10424, 11336, 12093

Offset: 0

Gus Wiseman, May 29 2019

nonn

The number of submultisets of a partition is the product of its multiplicities, each plus one.

The Heinz numbers of these partitions are given by A325798.

			The a(2) = 1 through a(9) = 14 partitions:
  (2)  (3)  (4)   (5)   (6)    (7)    (8)     (9)
            (22)  (32)  (33)   (43)   (44)    (54)
            (31)  (41)  (42)   (52)   (53)    (63)
                        (51)   (61)   (62)    (72)
                        (222)  (322)  (71)    (81)
                        (411)  (331)  (332)   (333)
                               (511)  (422)   (432)
                                      (431)   (441)
                                      (521)   (522)
                                      (611)   (531)
                                      (2222)  (621)
                                      (5111)  (711)
                                              (3222)
                                              (6111)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A002033, A088880, A088881, A108917, A126796, A307699, A325694, A325792, A325798, A325828, A325830, A325831, A325832, A325833, A325836.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
      (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
    end:
a:= n-> add(b(n$2, k), k=0..n):
seq(a(n), n=0..55);  # Alois P. Heinz, Aug 17 2019

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])<=n&]],{n,0,30}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, Function[w, b[w, Min[w, i - 1], Quotient[p, j + 1]]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := Sum[b[n, n, k], {k, 0, n}];
a /@ Range[0, 55] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

a(n) = A000041(n) - A325831(n).

For n even, A325833(n) = a(n) - A325830(n/2); for n odd, A325833(n) = a(n).

A088881 If A056239(m) = n, then a(n) is the maximum value of A000005(m).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 20, 24, 30, 36, 42, 48, 60, 72, 84, 96, 112, 128, 144, 168, 192, 224, 256, 288, 336, 384, 432, 480, 540, 600, 672, 768, 864, 960, 1080, 1200, 1320, 1440, 1620, 1800, 1980, 2160, 2400, 2640, 2880, 3240, 3600, 3960, 4320, 4800, 5280

Offset: 0

Naohiro Nomoto, Nov 28 2003

Maximum number of submultisets among all integer partitions of n. - Gus Wiseman, Jun 30 2019

			The partition (3,2,1,1,1) has 16 submultisets, which is more than for any other partition of 8, so a(8) = 16. - _Gus Wiseman_, Jun 30 2019

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

Cf. A088880, A307699, A325694, A325798, A325828, A325830, A325831, A325832, A325833, A325834.

b:= proc(n, i) option remember; `if`(n=0 or i<2, n+1,
       max(seq((j+1)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n, n):
seq (a(n), n=0..100);  # Alois P. Heinz, Aug 09 2012

$RecursionLimit = 1000; b[n_, i_] :=  b[n, i] = If[n == 0 || i<2, n+1, Max[Table[ (j+1)*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table [a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 15 2015, after Alois P. Heinz *)
Table[Max@@(Times@@(1+Length/@Split[#])&)/@IntegerPartitions[n],{n,0,30}] (* Gus Wiseman, Jun 30 2019 *)

A325832 Number of integer partitions of n whose number of submultisets is greater than or equal to n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 8, 13, 16, 22, 35, 50, 58, 85, 120, 162, 199, 267, 347, 462, 592, 773, 1006, 1293, 1504, 1929, 2455, 3081, 3859, 4815, 5953, 7363, 8737, 10743, 13193, 16102, 19241, 23413, 28344, 34260, 40911, 49197, 58917, 70515, 84055, 100070, 118914

Offset: 0

Gus Wiseman, May 25 2019

nonn

The number of submultisets of a partition is the product of its multiplicities, each plus one.

The Heinz numbers of these partitions are given by A325796.

			The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)   (21)   (31)    (221)    (321)     (421)      (431)
       (11)  (111)  (211)   (311)    (411)     (2221)     (521)
                    (1111)  (2111)   (2211)    (3211)     (3221)
                            (11111)  (3111)    (4111)     (3311)
                                     (21111)   (22111)    (4211)
                                     (111111)  (31111)    (5111)
                                               (211111)   (22211)
                                               (1111111)  (32111)
                                                          (41111)
                                                          (221111)
                                                          (311111)
                                                          (2111111)
                                                          (11111111)

Alois P. Heinz, Table of n, a(n) for n = 0..500

Cf. A002033, A098859, A108917, A126796, A325694, A325792, A325796, A325828, A325830, A325831, A325833, A325834, A325836.

b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
      (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
    end:
a:= n-> combinat[numbpart](n)-add(b(n$2, k), k=0..n-1):
seq(a(n), n=0..55);  # Alois P. Heinz, Aug 17 2019

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])>=n&]],{n,0,30}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, Function [w, b[w, Min[w, i - 1], p/(j + 1)]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := PartitionsP[n] - Sum[b[n, n, k], {k, 0, n - 1}];
Table[a[n], {n, 0, 55}] (* Jean-François Alcover, May 16 2021, after Alois P. Heinz *)

a(n) = A000041(n) - A325833(n).

For n even, a(n) = A325831(n) + A325830(n/2); for n odd, a(n) = A325831(n).

A307699 Numbers k such that there is no integer partition of k with exactly k-1 submultisets.

Original entry on oeis.org

0, 1, 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32, 38, 42, 44, 48, 50, 54, 60, 62, 66, 68, 72, 74, 80, 84, 86, 90, 92, 98, 102, 104, 108, 110, 114, 122, 126, 128, 132, 134, 138, 140, 146, 150, 152, 158, 164, 168, 170, 174, 180, 182, 186, 192, 194, 198, 200, 206

Offset: 1

Gus Wiseman, May 30 2019

nonn

After a(1) = 0, first differs from A229488 in lacking 56.
The number of submultisets of a partition is the product of its multiplicities, each plus one.
{a(n)-1} contains all odd numbers m = p*q*... such that gcd(p-1, q-1, ...) > 2. In particular, {a(n)-1} contains all powers of all primes > 3. Proof: If g is the greatest common divisor, then all factors of k are congruent to 1 modulo g, and thus all multiplicities of any valid multiset are divisible by g. However, the required sum is congruent to 2 modulo g, and so no such multiset can exist. - Charlie Neder, Jun 06 2019

			The sequence of positive terms together with their prime indices begins:
   1: {}
   2: {1}
   6: {1,2}
   8: {1,1,1}
  12: {1,1,2}
  14: {1,4}
  18: {1,2,2}
  20: {1,1,3}
  24: {1,1,1,2}
  26: {1,6}
  30: {1,2,3}
  32: {1,1,1,1,1}
  38: {1,8}
  42: {1,2,4}
  44: {1,1,5}
  48: {1,1,1,1,2}
  50: {1,3,3}
  54: {1,2,2,2}
  60: {1,1,2,3}
Partitions realizing the desired number of submultisets for each non-term are:
   3: (3)
   4: (22)
   5: (41)
   7: (511)
   9: (621)
  10: (4411)
  11: (71111)
  13: (9211)
  15: (9111111)
  16: (661111)
  17: (9521)
  19: (94411)
  21: (981111)
  22: (88111111)
  23: (32222222222)
  25: (99421)
  27: (3222222222222)
  28: (994411)
  29: (98222222)

Positions of zeros in A325836.

Cf. A002033, A088880, A088881, A098859, A108917, A126796, A276024, A325694, A325792, A325798, A325828, A325830, A325833, A325834, A325835.

Select[Range[50],Function[n,Select[IntegerPartitions[n], Times@@(1+Length/@Split[#])==n-1&]=={}]]

More terms from Alois P. Heinz, May 30 2019

A325835 Number of integer partitions of 2*n having one more distinct submultiset than distinct subset-sums.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 9, 10, 14, 22, 30, 33, 46, 52, 74, 107, 101, 123, 171, 182, 225

Offset: 0

Gus Wiseman, May 29 2019

The number of submultisets of a partition is the product of its multiplicities, each plus one. A subset-sum of an integer partition is the sum of some submultiset of its parts. These are partitions with one subset-sum which is the sum of two distinct submultisets, while all others are the sum of only one submultiset.

The Heinz numbers of these partitions are given by A325802.

			The a(2) = 1 through a(8) = 14 partitions:
  (211)  (321)   (422)    (532)     (633)      (743)       (844)
         (3111)  (431)    (541)     (642)      (752)       (853)
                 (41111)  (5221)    (651)      (761)       (862)
                          (5311)    (4332)     (7322)      (871)
                          (511111)  (5331)     (7331)      (5443)
                                    (6222)     (7421)      (7441)
                                    (6411)     (7511)      (7531)
                                    (33222)    (72221)     (8332)
                                    (6111111)  (74111)     (8521)
                                               (71111111)  (8611)
                                                           (82222)
                                                           (83311)
                                                           (85111)
                                                           (811111111)
For example, the partition (7,5,3,1) has submultisets (), (1), (3), (5), (7), (3,1), (5,1), (5,3), (7,1), (7,3), (7,5), (5,3,1), (7,3,1), (7,5,1), (7,5,3), (7,5,3,1), all of which have different sums except for (5,3) and (7,1), which both sum to 8, so (7,5,3,1) is counted under a(8).

Cf. A002033, A088880, A088881, A108917, A126796, A276024, A325799, A325800, A325801, A325802, A325828, A325830, A325833, A325836.

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])==1+Length[Union[Total/@Subsets[#]]]&]],{n,0,20,2}]

cp's OEIS Frontend

A325792 Positive integers with as many proper divisors as the sum of their prime indices.

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A325833 Number of integer partitions of n whose number of submultisets is less than n.

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A325830 Number of integer partitions of 2*n having exactly 2*n submultisets.

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A325836 Number of integer partitions of n having n - 1 different submultisets.

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A325831 Number of integer partitions of n whose number of submultisets is greater than n.

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A325834 Number of integer partitions of n whose number of submultisets is less than or equal to n.

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A088881 If A056239(m) = n, then a(n) is the maximum value of A000005(m).

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A325830 Number of integer partitions of 2n having exactly 2n submultisets.