A325834 -id:A325834 - cp's OEIS frontend

A325798 Numbers with at most as many divisors as the sum of their prime indices.

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89

Offset: 1

Gus Wiseman, May 23 2019

nonn

First differs from the complement of A325781 in lacking 156.

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 sequence of terms together with their prime indices begins:
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  27: {2,2,2}
  28: {1,1,4}
  29: {10}
  31: {11}

Positions of nonpositive terms in A325794.
Heinz numbers of the partitions counted by A325834.
Cf. A000005, A002033, A056239, A112798, A299702.
Cf. A325694, A325780, A325781, A325792, A325793, A325795, A325796, A325797.

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

A325828 Number of integer partitions of n having exactly n + 1 submultisets.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 3, 1, 12, 1, 3, 4, 21, 1, 14, 1, 18, 4, 3, 1, 116, 3, 3, 12, 25, 1, 40, 1, 271, 4, 3, 4, 325, 1, 3, 4, 295, 1, 56, 1, 36, 47, 3, 1, 3128, 4, 32, 4, 44, 1, 407, 4, 566, 4, 3, 1, 1598, 1, 3, 65, 10656, 5, 90, 1, 54, 4, 84, 1

Offset: 0

Gus Wiseman, May 25 2019

nonn

The Heinz numbers of these partitions are given by A325792.

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

			The 12 = 11 + 1 submultisets of the partition (4331) are: (), (1), (3), (4), (31), (33), (41), (43), (331), (431), (433), (4331), so (4331) is counted under a(11).
The a(5) = 3 through a(11) = 12 partitions:
  221    111111  421      3311      22221      1111111111  4322
  311            2221     11111111  51111                  4331
  11111          4111               111111111              4421
                 1111111                                   5411
                                                           6221
                                                           6311
                                                           7211
                                                           33311
                                                           44111
                                                           222221
                                                           611111
                                                           11111111111

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

Cf. A002033, A098859, A126796, A188431, A325694, A325792, A325793, A325830, 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-> 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[#])-1==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 = Quotient[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 11 2021, after Alois P. Heinz *)

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

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).

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

A325987 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with k submultisets, k > 0.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 1, 1, 1, 0, 1, 0, 2, 0, 3, 0, 1, 0, 1, 1, 3, 0, 1, 1, 2, 1, 1, 0, 1, 0, 3, 0, 3, 0, 4, 0, 1, 0, 3, 0, 1, 1, 3, 1, 3, 0, 3, 2, 1, 0, 4, 0, 1, 1, 1, 0, 1, 0, 5, 0, 3, 0, 5, 0, 3, 0, 6, 0, 1, 0, 3, 0, 2, 0, 1, 0, 1, 1, 4, 0

Offset: 0

Gus Wiseman, May 30 2019

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

			Triangle begins:
  1
  0 1
  0 1 1
  0 1 0 2
  0 1 1 1 1 1
  0 1 0 2 0 3 0 1
  0 1 1 3 0 1 1 2 1 1
  0 1 0 3 0 3 0 4 0 1 0 3
  0 1 1 3 1 3 0 3 2 1 0 4 0 1 1 1
  0 1 0 5 0 3 0 5 0 3 0 6 0 1 0 3 0 2 0 1
  0 1 1 4 0 5 0 7 2 1 1 4 0 1 2 5 0 3 0 2 1 0 0 2
Row n = 7 counts the following partitions (empty columns not shown):
  (7)  (43)  (322)  (421)      (31111)  (3211)
       (52)  (331)  (2221)              (22111)
       (61)  (511)  (4111)              (211111)
                    (1111111)

Alois P. Heinz, Rows n = 0..60, flattened

Row lengths are A088881.
Row sums are A000041.
Diagonal n = k is A325830 interspersed with zeros.
Diagonal n + 1 = k is A325828.
Diagonal n - 1 = k is A325836.
Column k = 3 appears to be A137719.
Cf. A000005, A000712, A002033, A005179, A088880, A108917, A126796.
Cf. A325694, A325792, A325793, A325831, A325832, A325833, A325834.

Table[Length[Select[IntegerPartitions[n],Times@@(1+Length/@Split[#])==k&]],{n,0,10},{k,1,Max@@(Times@@(1+Length/@Split[#])&)/@IntegerPartitions[n]}]

Sum_{k=1..A088881(n)} k * T(n,k) = A000712(n). - Alois P. Heinz, Aug 17 2019

cp's OEIS Frontend

A325798 Numbers with at most as many divisors as the sum of their prime indices.

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A325828 Number of integer partitions of n having exactly n + 1 submultisets.

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

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

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

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Maple

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