A325835 -id:A325835 - cp's OEIS frontend

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

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 *)

A325802 Numbers with one more divisor than distinct subset-sums of their prime indices.

Original entry on oeis.org

12, 30, 40, 63, 70, 112, 154, 165, 198, 220, 273, 286, 325, 351, 352, 364, 442, 525, 550, 561, 595, 646, 675, 714, 741, 748, 765, 832, 850, 874, 918, 931, 952, 988, 1045, 1173, 1254, 1334, 1425, 1495, 1539, 1564, 1653, 1666, 1672, 1771, 1794, 1798, 1870, 1900

Offset: 1

Gus Wiseman, May 23 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. A subset-sum of an integer partition is any sum of a submultiset of it.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so these are Heinz numbers of the partitions counted by A325835.

			The sequence of terms together with their prime indices begins:
   12: {1,1,2}
   30: {1,2,3}
   40: {1,1,1,3}
   63: {2,2,4}
   70: {1,3,4}
  112: {1,1,1,1,4}
  154: {1,4,5}
  165: {2,3,5}
  198: {1,2,2,5}
  220: {1,1,3,5}
  273: {2,4,6}
  286: {1,5,6}
  325: {3,3,6}
  351: {2,2,2,6}
  352: {1,1,1,1,1,5}
  364: {1,1,4,6}
  442: {1,6,7}
  525: {2,3,3,4}
  550: {1,3,3,5}
  561: {2,5,7}

Robert Israel, Table of n, a(n) for n = 1..10000

Positions of 1's in A325801.
Cf. A000005, A056239, A108917, A112798, A276024, A299701, A299702.
Cf. A325694, A325780, A325781, A325799, A325800, A325835.

filter:= proc(n) local F,t,S,i;
  F:= map(t -> [numtheory:-pi(t[1]),t[2]], ifactors(n)[2]);
  S:= {0}:
  for t in F do
   S:= map(s -> seq(s + i*t[1],i=0..t[2]),S);
  od;
  nops(S) = mul(t[2]+1,t=F)-1
end proc:
select(filter, [$1..2000]); # Robert Israel, Oct 30 2024

Select[Range[100],DivisorSigma[0,#]==1+Length[Union[hwt/@Divisors[#]]]&]

A000005(a(n)) = 1 + A299701(a(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

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

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A325802 Numbers with one more divisor than distinct subset-sums of their prime indices.

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

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