A318361 -id:A318361 - cp's OEIS frontend

A318286 Number of strict multiset partitions of a multiset whose multiplicities are the prime indices of n.

1, 1, 1, 2, 2, 3, 2, 5, 5, 5, 3, 9, 4, 7, 9, 15, 5, 18, 6, 16, 14, 10, 8, 31, 17, 14, 40, 25, 10, 34, 12, 52, 21, 19, 27, 70, 15, 25, 31, 59, 18, 57, 22, 38, 80, 33, 27, 120, 46, 67, 44, 56, 32, 172, 42, 100, 61, 43, 38, 141, 46, 55, 143, 203, 64, 91, 54, 80

Offset: 1

Gus Wiseman, Aug 23 2018

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A001055, A007716, A045778, A181821, A305936, A316980, A317776.

Cf. A318283, A318284, A318285, A318287, A318360, A318361.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
Table[Length[strfacs[Times@@Prime/@nrmptn[n]]],{n,60}]

permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i, 2], j, primepi(f[i, 1]))))}
count(sig)={my(r=0, A=O(x*x^vecmax(sig))); for(n=1, vecsum(sig)+1, my(s=0); forpart(p=n, my(q=1/prod(i=1, #p, 1 - x^p[i] + A)); s+=prod(i=1, #sig, polcoef(q, sig[i]))*(-1)^#p*permcount(p)); r+=(-1)^n*s/n!); r/2}
a(n)={if(n==1, 1, count(sig(n)))} \\ Andrew Howroyd, Dec 18 2018

a(n) = A045778(A181821(n)).

a(prime(n)^k) = A219585(n, k). - Andrew Howroyd, Dec 17 2018

A383533 Number of integer partitions of n with no ones such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 4, 5, 8, 8, 11, 13, 17, 22, 25, 30, 37, 44, 53, 69, 77, 93, 111, 130, 153, 181, 220, 249, 295

Offset: 0

Gus Wiseman, May 07 2025

The Heinz numbers of these partitions are the odd terms of A382913.

Also the number of integer partitions y of n with no ones such that the normal multiset (in which i appears y_i times) is a Look-and-Say partition.

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is counted under a(6).
The a(2) = 1 through a(10) = 8 partitions:
  (2)  (3)  (4)  (5)    (6)    (7)    (8)    (9)      (10)
                 (3,2)  (3,3)  (4,3)  (4,4)  (5,4)    (5,5)
                        (4,2)  (5,2)  (5,3)  (6,3)    (6,4)
                                      (6,2)  (7,2)    (7,3)
                                             (4,3,2)  (8,2)
                                                      (4,3,3)
                                                      (4,4,2)
                                                      (5,3,2)

The number of such families is A383706.
Allowing ones gives A383708 (ranks A382913), complement A383710 (ranks A382912).
The complement is counted by A383711.
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
Cf. A044813, A047966, A089259, A116540, A130091, A317141, A318361, A381441, A381454, A383013.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y], UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n], FreeQ[#,1]&&!pof[#]=={}&]],{n,0,15}]

A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 23, 48, 101, 208, 434

Offset: 0

Gus Wiseman, Mar 29 2025

We call a multiset normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The multiset {1,2,2,3,3} can be partitioned into a set of sets with distinct sums in 4 ways:
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
so is counted under a(5).
The multisets counted by A382214 but not by A382216 are:
  {1,1,1,1,2,2,3,3,3}
  {1,1,2,2,2,2,3,3,3}
The a(1) = 1 through a(5) = 11 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
              {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
                       {1,2,3,3}  {1,1,2,3,4}
                       {1,2,3,4}  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, A116539.
The complement is counted by A382202.
Without distinct sums we have A382214, complement A292432.
The case of a unique choice is counted by A382459, without distinct sums A382458.
For Heinz numbers: A293243, A381806, A382075, A382200.
For integer partitions: A381990, A381992, A382077, A382078.
Strong version: A382523, A382430, A381996, A292444.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A000670, A007716, A050320, A116540, A255903, A275780, A317532, A326518, A326519, A382429, A382460.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,5}]

A383711 Number of integer partitions of n with no ones such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 1, 3, 3, 4, 6, 10, 11, 17, 19, 30, 36, 51, 61, 84, 96, 133, 160, 209, 253, 325, 393, 488, 598, 744

Offset: 0

Gus Wiseman, May 07 2025

The Heinz numbers of these partitions are the odd terms of A382912.

Also the number of integer partitions of n with no ones whose normal multiset (in which i appears y_i times) is not a Look-and-Say partition.

			For y = (3,3) we can choose disjoint strict partitions ((2,1),(3)), so (3,3) is not counted under a(6).
The a(4) = 1 through a(12) = 10 partitions:
  (22)  .  (222)  (322)  (332)   (333)   (622)    (443)    (444)
                         (422)   (522)   (3322)   (722)    (822)
                         (2222)  (3222)  (4222)   (3332)   (3333)
                                         (22222)  (4322)   (4332)
                                                  (5222)   (4422)
                                                  (32222)  (5322)
                                                           (6222)
                                                           (33222)
                                                           (42222)
                                                           (222222)

The complement without ones is counted by A383533.
The number of these families is A383706.
Allowing ones gives A383710 (ranks A382912), complement A383708 (ranks A382913).
A048767 is the Look-and-Say transform, fixed points A048768 (counted by A217605).
A098859 counts partitions with distinct multiplicities, compositions A242882.
A239455 counts Look-and-Say or section-sum partitions, ranks A351294 or A381432.
A351293 counts non-Look-and-Say or non-section-sum partitions, ranks A351295 or A381433.
Cf. A044813, A047966, A089259, A116540, A130091, A317141, A318361, A381441, A381454, A383013.

pof[y_]:=Select[Join@@@Tuples[IntegerPartitions/@y],UnsameQ@@#&];
Table[Length[Select[IntegerPartitions[n],FreeQ[#,1]&&pof[#]=={}&]],{n,0,15}]

A188445 T(n,k) is the number of (n*k) X k binary arrays with nonzero rows in decreasing order and n ones in every column.

Original entry on oeis.org

1, 2, 0, 5, 1, 0, 15, 8, 0, 0, 52, 80, 5, 0, 0, 203, 1088, 205, 1, 0, 0, 877, 19232, 11301, 278, 0, 0, 0, 4140, 424400, 904580, 67198, 205, 0, 0, 0, 21147, 11361786, 101173251, 24537905, 250735, 80, 0, 0, 0, 115975, 361058000, 15207243828, 13744869502

Offset: 1

R. H. Hardin, Mar 31 2011

			Array begins:
============================================================================
n\k| 1 2 3   4       5          6             7              8             9
---+------------------------------------------------------------------------
1  | 1 2 5  15      52        203           877           4140         21147
2  | 0 1 8  80    1088      19232        424400       11361786     361058000
3  | 0 0 5 205   11301     904580     101173251    15207243828 2975725761202
4  | 0 0 1 278   67198   24537905   13744869502 11385203921707 ...
5  | 0 0 0 205  250735  425677958 1184910460297 ...
6  | 0 0 0  80  621348 5064948309 ...
7  | 0 0 0  15 1058139 ...
8  | 0 0 0   1 ...
...
Some solutions for 16 X 4:
  1 1 1 0    1 1 1 1    1 1 1 1    1 1 1 0    1 1 1 1
  1 0 1 1    1 1 0 1    1 1 0 0    1 0 1 1    1 1 0 0
  1 0 1 0    1 0 1 1    1 0 1 1    1 0 0 1    1 0 1 1
  1 0 0 1    1 0 0 0    1 0 0 0    1 0 0 0    1 0 0 0
  0 1 1 1    0 1 1 0    0 1 1 1    0 1 1 0    0 1 1 1
  0 1 0 1    0 1 0 0    0 1 0 0    0 1 0 1    0 1 0 0
  0 1 0 0    0 0 1 1    0 0 1 1    0 1 0 0    0 0 1 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 1 1    0 0 0 1
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0
  0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0    0 0 0 0

Andrew Howroyd, Table of n, a(n) for n = 1..181 (terms 1..69 from R. H. Hardin)

Rows 1..5 are A000110, A002718, A060486, A188446, A188447.
Columns 5..6 are A331127, A331129.
Column sums are A319190.
Cf. A059443, A060487, A188392, A219585, A318361, A330942, A330964, A331039.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); WeighT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}
T(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(1+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])/2)} \\ Andrew Howroyd, Dec 16 2018

A(n,k) = 0 for n > 2^(k-1). - Andrew Howroyd, Jan 24 2020

A381436 Irregular triangle read by rows where row k is the section-sum partition of the prime indices of n.

Original entry on oeis.org

1, 2, 1, 1, 3, 3, 4, 1, 1, 1, 2, 2, 4, 5, 3, 1, 6, 5, 5, 1, 1, 1, 1, 7, 3, 2, 8, 4, 1, 6, 6, 9, 3, 1, 1, 3, 3, 7, 2, 2, 2, 5, 1, 10, 6, 11, 1, 1, 1, 1, 1, 7, 8, 7, 3, 3, 12, 9, 8, 4, 1, 1, 13, 7, 14, 6, 1, 5, 2, 10, 15, 3, 1, 1, 1, 4, 4, 4, 3, 9, 7, 1, 16, 3, 2, 2

Offset: 1

Gus Wiseman, Feb 28 2025

Row-lengths are A051903.
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 section-sum partition of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The prime indices of 24 are (2,1,1,1), with sections ((2,1),(1),(1)), so row 24 is (3,1,1).
Triangle begins:
   1: (empty)
   2: 1
   3: 2
   4: 1 1
   5: 3
   6: 3
   7: 4
   8: 1 1 1
   9: 2 2
  10: 4
  11: 5
  12: 3 1
  13: 6
  14: 5
  15: 5
  16: 1 1 1 1

Row-lengths are A051903.
Row sums are A056239.
First part in each row is A066328.
Taking length instead of sum gives A238744, Heinz numbers A238745, conjugate A181819.
Partitions of this type are counted by A239455, complement A351293.
Heinz numbers are A381431 (union A381432, complement A381433, fixed A000961, A000005).
Rows appearing only once have Heinz numbers A381434, more than once A381435.
Last part in each row is A381437, counted by A381438.
The conjugate is A381440, Heinz numbers A048767 (union A351294, complement A351295).
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A001222, A003557, A005117, A047966, A116861, A130091, A380955.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[egs[prix[n]],{n,100}]

A381440 Irregular triangle read by rows where row k is the Look-and-Say partition of the prime indices of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 28 2025

Row lengths are A066328.
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 Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.

			The prime indices of 24 are (2,1,1,1), with Look-and-Say partition (3,1,1), so row 24 is (3,1,1).
The prime indices of 36 are (2,2,1,1), with Look-and-Say partition (2,2,2), so row 36 is (2,2,2).
Triangle begins:
   1: (empty)
   2: 1
   3: 1 1
   4: 2
   5: 1 1 1
   6: 1 1 1
   7: 1 1 1 1
   8: 3
   9: 2 2
  10: 1 1 1 1
  11: 1 1 1 1 1
  12: 2 1 1
  13: 1 1 1 1 1 1
  14: 1 1 1 1 1
  15: 1 1 1 1 1
  16: 4
  17: 1 1 1 1 1 1 1
  18: 2 2 1
  19: 1 1 1 1 1 1 1 1

Heinz numbers are A048767 (union A351294, complement A351295, fixed A048768, A217605).
First part in each row is A051903, conjugate A066328.
Last part in each row is A051904, conjugate A381437 (counted by A381438).
Row sums are A056239.
Row lengths are A066328.
Partitions of this type are counted by A239455, complement A351293.
The conjugate is A381436, Heinz numbers A381431 (union A381432, complement A381433).
Rows appearing only once have Heinz numbers A381540, more than once A381541.
A000040 lists the primes.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A047966, A071178, A091602, A116861, A130091, A181819, A212166, A238744, A380955.

Table[Sort[Join@@Cases[FactorInteger[n],{p_,k_}:>ConstantArray[k,PrimePi[p]]]]//Reverse,{n,30}]

A381437 Last part of the section-sum partition of the prime indices of n.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 4, 1, 2, 4, 5, 1, 6, 5, 5, 1, 7, 2, 8, 1, 6, 6, 9, 1, 3, 7, 2, 1, 10, 6, 11, 1, 7, 8, 7, 3, 12, 9, 8, 1, 13, 7, 14, 1, 2, 10, 15, 1, 4, 3, 9, 1, 16, 2, 8, 1, 10, 11, 17, 1, 18, 12, 2, 1, 9, 8, 19, 1, 11, 8, 20, 1, 21, 13, 3, 1, 9, 9, 22, 1, 2

Offset: 1

Gus Wiseman, Feb 28 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.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The prime indices of 972 are {1,1,2,2,2,2,2}, with section-sum partition (3,3,2,2,2), so a(972) = 2.

Positions of first appearances are A008578.
The length of this partition is A051903.
The conjugate version is A051904.
For first instead of last part we get A066328.
These partitions are counted by A239455, complement A351293.
Positions of 1 are A360013, complement A381439.
This is the least prime index of A381431 (see A381432, A381433, A381434, A381435).
This is the last part of row n of A381436 (see A381440, A048767, A351294, A351295).
Counting partitions by this statistic gives A381438.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Cf. A000720, A001221, A001222, A003557, A047966, A116861, A130091, A181819, A380955.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[If[n==1,0,Last[egs[prix[n]]]],{n,100}]

a(n) = A055396(A381431(n)).

A382214 Number of normal multisets of size n that can be partitioned into a set of sets.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 23, 48, 101, 210, 436, 894

Offset: 0

Gus Wiseman, Mar 29 2025

First differs from A382216 at a(9) = 210, A382216(9) = 208.

We call a multiset or multiset partition normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The normal multiset {1,1,1,1,2,2,3,3,3} has partition {{1},{3},{1,2},{1,3},{1,2,3}}, so is counted under a(9).
The a(1) = 1 through a(5) = 11 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
              {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
                       {1,2,3,3}  {1,1,2,3,4}
                       {1,2,3,4}  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}

Factorizations of this type are counted by A050326, distinct sums A381633.
Normal multiset partitions of this type are counted by A116539, distinct sums A381718.
The complement is counted by A292432.
Twice-partitions of this type are counted by A358914, distinct sums A279785.
The strong version is A381996, complement A292444.
For integer partitions we have A382077, ranks A382200, complement A382078, ranks A293243.
For distinct sums we have A382216, complement A382202.
The case of a unique choice is counted by A382458, distinct sums A382459.
A000670 counts patterns, ranked by A055932 and A333217, necklace A019536.
A001055 count factorizations, strict A045778.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A007716, A050320, A116540, A255903, A317532, A326517, A326519, A381990, A381992, A382428, A382460.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]] /@ Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]& /@ sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Select[mps[#], UnsameQ@@#&&And@@UnsameQ@@@#&]!={}&]],{n,0,5}]

A381435 Numbers appearing more than once in A381431 (section-sum partition of prime indices).

Original entry on oeis.org

5, 7, 11, 13, 17, 19, 23, 25, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 49, 51, 52, 53, 57, 58, 59, 61, 62, 65, 67, 68, 69, 71, 73, 74, 76, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 101, 103, 104, 106, 107, 109, 111, 113, 115, 116, 117, 118, 119

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
   5: {3}
   7: {4}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  26: {1,6}
  29: {10}
  31: {11}
  34: {1,7}
  37: {12}
  38: {1,8}
  39: {2,6}
  41: {13}
  43: {14}
  46: {1,9}
  47: {15}
  49: {4,4}
  51: {2,7}
  52: {1,1,6}

In A381431:
- fixed points are A000961, A000005
- conjugate is A048767, fixed points A048768, A217605
- all numbers present are A381432, conjugate A351294
- numbers missing are A381433, conjugate A351295
- numbers appearing only once are A381434, conjugate A381540
- numbers appearing more than once are A381435 (this), conjugate A381541
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.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions, complement A351293.
A381436 lists section-sum partition of prime indices, conjugate A381440.
Set multipartitions: A050320, A089259, A116540, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A051903, A066328, A116861, A130091, A212166, A239964, A317081, A381437.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Select[Range[100],Count[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]],#]>1&]

The complement is A381434 U A381433.

cp's OEIS Frontend

A318286 Number of strict multiset partitions of a multiset whose multiplicities are the prime indices of n.

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A383533 Number of integer partitions of n with no ones such that it is possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

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A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

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A383711 Number of integer partitions of n with no ones such that it is not possible to choose a family of pairwise disjoint strict integer partitions, one of each part.

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A188445 T(n,k) is the number of (n*k) X k binary arrays with nonzero rows in decreasing order and n ones in every column.

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A381436 Irregular triangle read by rows where row k is the section-sum partition of the prime indices of n.

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A381440 Irregular triangle read by rows where row k is the Look-and-Say partition of the prime indices of n.

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A381437 Last part of the section-sum partition of the prime indices of n.

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A382214 Number of normal multisets of size n that can be partitioned into a set of sets.

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