A300383 -id:A300383 - cp's OEIS frontend

A321471 Heinz numbers of integer partitions that can be partitioned into blocks with sums {1, 2, ..., k} for some k.

2, 6, 8, 30, 36, 40, 48, 64, 210, 252, 270, 280, 300, 324, 336, 360, 400, 432, 448, 480, 576, 640, 768, 1024, 2310, 2772, 2940, 2970, 3080, 3150, 3300, 3528, 3564, 3696, 3780, 3920, 3960, 4050, 4200, 4400, 4500, 4536, 4704, 4752, 4860, 4928, 5040, 5280, 5400

Offset: 1

Gus Wiseman, Nov 13 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

These partitions are those that are finer than (k, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + k, for some k, ordered by refinement.

			The sequence of all integer partitions whose Heinz numbers are in the sequence begins: (1), (21), (111), (321), (2211), (3111), (21111), (111111), (4321), (42211), (32221), (43111), (33211), (222211), (421111), (322111), (331111), (2221111), (4111111), (3211111), (22111111), (31111111), (211111111), (1111111111).
The partition (322111) has Heinz number 360 and can be partitioned as ((1)(2)(3)(112)), ((1)(2)(12)(13)), or ((1)(11)(3)(22)), so 360 belongs to the sequence.

Subsequence of A242422.
Cf. A001970, A002846, A056239, A066723, A112798, A213427, A242422, A265947, A300383, A317141.
Cf. A321467, A321468, A321470, A321472, A321514.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[2,1000],Select[Map[Total[primeMS[#]]&,facs[#],{2}],Sort[#]==Range[Max@@#]&]!={}&]

A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Gus Wiseman, Mar 22 2025

nonn

Also numbers with a unique factorization into prime powers with distinct sums of prime indices.

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, sum A056239.

			The prime indices of 270 are {1,2,2,2,3}, and there are two multiset partitions into constant multisets with distinct sums: {{1},{2},{3},{2,2}} and {{1},{3},{2,2,2}}, so 270 is not in the sequence.
The prime indices of 300 are {1,1,2,3,3}, of which there are no multiset partitions into constant multisets with distinct sums, so 300 is not in the sequence.
The prime indices of 360 are {1,1,1,2,2,3}, of which there is only one multiset partition into constant multisets with distinct sums: {{1},{1,1},{3},{2,2}}, so 360 is in the sequence.
The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    6: {1,2}
    7: {4}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   17: {7}
   18: {1,2,2}
   19: {8}
   20: {1,1,3}
   21: {2,4}
   22: {1,5}
   23: {9}
   24: {1,1,1,2}
   25: {3,3}

For distinct blocks instead of block-sums we have A004709, counted by A000726.
Twice-partitions of this type are counted by A279786.
MM-numbers of these multiset partitions are A326535 /\ A355743.
These are the positions of 1 in A381635.
For no choices we have A381636 (zeros of A381635), counted by A381717.
For strict instead of constant blocks we have A381870, counted by A382079.
Partitions of this type (unique into constant with distinct) are counted by A382301.
Normal multiset partitions of this type are counted by A382203.
A001055 counts multiset partitions, see A317141 (upper), A300383 (lower), A265947.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000688, A000720, A001222, A003963, A005117, A047966, A050361, A293511, A300385, A381716.
Cf. A006171, A279784, A295935.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
pfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[pfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],PrimePowerQ]}]];
Select[Range[100],Length[Select[pfacs[#],UnsameQ@@hwt/@#&]]==1&]

A321472 Heinz numbers of integer partitions whose parts can be further partitioned and flattened to obtain the partition (k, ..., 3, 2, 1) for some k.

Original entry on oeis.org

2, 5, 6, 13, 21, 22, 25, 29, 30, 46, 47, 57, 73, 85, 86, 91, 102, 107, 121, 123, 130, 142, 147, 151, 154, 165, 175, 185, 197, 201, 206, 210, 217, 222, 257, 298, 299

Offset: 1

Gus Wiseman, Nov 13 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

These partitions are those that are coarser than (k, ..., 3, 2, 1) in the poset of integer partitions of 1 + 2 + ... + k, for some k, ordered by refinement.

			The sequence of all integer partitions whose Heinz numbers are in the sequence begins: (1), (3), (2,1), (6), (4,2), (5,1), (3,3), (10), (3,2,1), (9,1), (15), (8,2), (21), (7,3), (14,1), (6,4), (7,2,1), (28), (5,5), (13,2), (6,3,1), (20,1), (4,4,2), (36), (5,4,1), (5,3,2), (4,3,3), (12,3), (45), (19,2), (27,1), (4,3,2,1).

Subsequence of A242422.
Cf. A001970, A002846, A056239, A066723, A112798, A213427, A242422, A265947, A300383, A317141.
Cf. A321467, A321468, A321470, A321471, A321514.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[2,200],Select[Sort/@Join@@@Tuples[IntegerPartitions/@primeMS[#]],Sort[#]==Range[Max@@#]&]!={}&]

A321514 Number of ways to choose a factorization of each integer from 2 to n into factors > 1.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 4, 12, 24, 48, 48, 192, 192, 384, 768, 3840, 3840, 15360, 15360, 61440, 122880, 245760, 245760, 1720320, 3440640, 6881280, 20643840, 82575360, 82575360, 412876800, 412876800, 2890137600, 5780275200, 11560550400, 23121100800, 208089907200

Offset: 1

Gus Wiseman, Nov 11 2018

nonn

			The a(8) = 12 ways to choose a factorization of each integer from 2 to 8:
  (2)*(3)*(4)*(5)*(6)*(7)*(8)
  (2)*(3)*(4)*(5)*(6)*(7)*(2*4)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(8)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(8)
  (2)*(3)*(4)*(5)*(6)*(7)*(2*2*2)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(8)
  (2)*(3)*(4)*(5)*(2*3)*(7)*(2*2*2)
  (2)*(3)*(2*2)*(5)*(6)*(7)*(2*2*2)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*4)
  (2)*(3)*(2*2)*(5)*(2*3)*(7)*(2*2*2)

Dominates A321468.

Cf. A001055, A050336, A066723, A076716, A157612, A281113, A300383, A317144, A317145, A321467, A321470, A321471, A321472.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Array[Length[facs[#]]&,n,1,Times],{n,30}]

a(n) = Product_{k = 1..n} A001055(k).

A381637 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into blocks with distinct sums.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Mar 10 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 prime indices of 84 are {1,1,2,4}, with 7 multiset partitions into blocks with distinct sums:
  {{1,1,2,4}}
  {{1},{1,2,4}}
  {{2},{1,1,4}}
  {{1,1},{2,4}}
  {{1,2},{1,4}}
  {{1},{2},{1,4}}
  {{1},{4},{1,2}}
with block-sums: {8}, {1,7}, {2,6}, {2,6}, {3,5}, {1,2,5}, {1,3,4}, of which 6 are distinct, so a(84) = 6.

Allowing any block-sums gives A317141  (lower A300383), before sums A001055.
Before taking sums we had A321469.
For distinct blocks instead of distinct block-sums we have A381452.
If each block is a set we have A381634 (zeros A381806), before sums A381633.
For equal instead of distinct block-sums we have A381872, before sums A321455.
Other multiset partitions of prime indices:
- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).
- For set multipartitions (A050320) see A381078 (upper), A381454 (lower).
- For sets of constant multisets (A050361) see A381715.
- For sets of constant multisets with distinct sums (A381635) see A381716, A381636.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A000720, A001222, A001970, A002846, A045778, A066328, A213385, A299200, A299201, A299202, A300385, A317142.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
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[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@Total/@#&]]],{n,100}]

A383014 Numbers whose prime indices can be partitioned into constant blocks with a common sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 36, 37, 40, 41, 43, 47, 48, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 108, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169

Offset: 1

Gus Wiseman, Apr 22 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, sum A056239.

			The prime indices of 36 are {1,1,2,2}, and a partition into constant blocks with a common sum is: {{2},{2},{1,1}}, so 36 is in the sequence.
The prime indices of 43200 are {1,1,1,1,1,1,2,2,2,3,3}, and a partition into constant blocks with a common sum is: {{{1,1,1,1,1,1},{2,2,2},{3,3}}}, so 43200 is in the sequence.
The prime indices of 520000 are {1,1,1,1,1,1,3,3,3,3,6} and a partition into constant blocks with a common sum is: {{1,1,1,1,1,1},{3,3},{3,3},{6}}, so 520000 is in the sequence.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  11: {5}
  12: {1,1,2}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  37: {12}
  40: {1,1,1,3}

Twice-partitions of this type (constant blocks with a common sum) are counted by A279789.
Includes all elements of A353833.
For distinct sums we have the complement of A381636.
For strict blocks we have the complement of A381719.
For distinct sums and strict blocks we have the complement of A381806.
The complement is A381871, counted by A381993.
These are the positions of positive terms in A381995.
Partitions of this type are counted by A383093.
Constant blocks: A000688, A006171, A279784, A295935, A381453 (lower), A381455 (upper).
A001055 counts factorizations (multiset partitions of prime indices), strict A045778.
A050361 counts factorizations into distinct prime powers.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000720, A000961, A001222, A321469, A381635, A381715, A381716, A381717.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
mce[y_]:=Table[ConstantArray[y[[1]],#]&/@ptn, {ptn,IntegerPartitions[Length[y]]}];
Select[Range[100], Select[Join@@@Tuples[mce/@Split[prix[#]]], SameQ@@Total/@#&]!={}&]

A301595 Number of thrice-partitions of n.

Original entry on oeis.org

1, 1, 4, 10, 34, 80, 254, 604, 1785, 4370, 11986, 29286, 80355, 193137, 505952, 1239348, 3181970, 7686199, 19520906, 46931241, 117334784, 282021070, 693721166, 1659075192, 4063164983, 9651686516, 23347635094, 55405326513, 133021397071, 313842472333, 749299686508

Offset: 0

Gus Wiseman, Mar 24 2018

nonn

A thrice-partition of n is a choice of a twice-partition of each part in a partition of n. Thrice-partitions correspond to intervals in the lattice form of the multiorder of integer partitions.

			The a(3) = 10 thrice-partitions:
  ((3)), ((21)), ((111)), ((2)(1)), ((11)(1)), ((1)(1)(1)),
  ((2))((1)), ((11))((1)), ((1)(1))((1)),
  ((1))((1))((1)).

Alois P. Heinz, Table of n, a(n) for n = 0..3244
Gus Wiseman, The a(4) = 34 thrice-partitions of 4.

Cf. A000041, A001383, A001970, A061260, A063834, A119442, A196545, A281113, A289501, A300383, A301422, A301462, A301480, A301595, A301598, A301706.

Column k=3 of A323718.

b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
      1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
    end:
a:= n-> b(n$2, 3):
seq(a(n), n=0..35);  # Alois P. Heinz, Jan 25 2019

twie[n_]:=Sum[Times@@PartitionsP/@ptn,{ptn,IntegerPartitions[n]}];
thrie[n_]:=Sum[Times@@twie/@ptn,{ptn,IntegerPartitions[n]}];
Array[thrie,30]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1,
     1, b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
a[n_] := b[n, n, 3];
a /@ Range[0, 35] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

O.g.f.: Product_{n > 0} 1/(1 - A063834(n) x^n).

a(0)=1 prepended by Alois P. Heinz, Jan 25 2019

A316092 Heinz numbers of integer partitions of prime numbers into prime parts.

Original entry on oeis.org

3, 5, 11, 15, 17, 31, 33, 41, 45, 59, 67, 83, 93, 109, 127, 153, 157, 177, 179, 191, 211, 241, 275, 277, 283, 297, 327, 331, 353, 367, 369, 375, 401, 405, 425, 431, 459, 461, 509, 537, 547, 563, 587, 599, 603, 605, 617, 709, 739, 773, 775, 797, 825, 831, 837

Offset: 1

Gus Wiseman, Jun 24 2018

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			Sequence of integer partitions of prime numbers into prime parts together with their Heinz numbers begins:
   3: (2)
   5: (3)
  11: (5)
  15: (2,3)
  17: (7)
  31: (11)
  33: (2,5)
  41: (13)
  45: (2,2,3)

Cf. A000041, A000607, A056239, A056768, A076610, A100118, A112798, A215366, A296150, A300383, A316091.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[900],And[PrimeQ[Total[primeMS[#]]],And@@PrimeQ/@primeMS[#]]&]

A317143 In the ranked poset of integer partitions ordered by refinement, row n lists the Heinz numbers of integer partitions finer (less) than or equal to the integer partition with Heinz number n.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 8, 6, 8, 7, 9, 10, 12, 16, 8, 9, 12, 16, 10, 12, 16, 11, 14, 15, 18, 20, 24, 32, 12, 16, 13, 21, 22, 25, 27, 28, 30, 36, 40, 48, 64, 14, 18, 20, 24, 32, 15, 18, 20, 24, 32, 16, 17, 26, 33, 35, 42, 44, 45, 50, 54, 56, 60, 72, 80, 96, 128

Offset: 1

Gus Wiseman, Jul 22 2018

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

If x and y are partitions of the same integer and it is possible to produce x by further partitioning the parts of y, flattening, and sorting, then x <= y.

			The partitions finer than or equal to (2,2) are (2,2), (2,1,1), (1,1,1,1), with Heinz numbers 9, 12, 16, so the 9th row is {9, 12, 16}.
Triangle begins:
   1
   2
   3   4
   4
   5   6   8
   6   8
   7   9  10  12  16
   8
   9  12  16
  10  12  16
  11  14  15  18  20  24  32
  12  16
  13  21  22  25  27  28  30  36  40  48  64
  14  18  20  24  32
  15  18  20  24  32
  16
  17  26  33  35  42  44  45  50  54  56  60  72  80  96 128

Row lengths are A300383.

Cf. A002846, A056239, A213427, A215366, A265947, A296150, A299201, A317141.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Union[Times@@@Map[Prime,Join@@@Tuples[IntegerPartitions/@primeMS[n]],{2}]],{n,12}]

A301764 Number of ways to choose a constant rooted partition of each part in a constant rooted partition of n such that the flattened sequence is also constant.

Original entry on oeis.org

1, 1, 2, 3, 4, 4, 6, 5, 6, 7, 8, 5, 10, 7, 8, 10, 10, 6, 12, 7, 12, 13, 10, 5, 14, 12, 11, 11, 14, 7, 18, 9, 12, 13, 11, 12, 20, 10, 10, 13, 18, 9, 20, 9, 14, 20, 12, 5, 20, 15, 19, 14, 17, 7, 18, 16, 20, 17, 12, 5, 26, 13, 12, 21, 18, 17, 24, 9, 15, 13, 22, 9

Offset: 1

Gus Wiseman, Mar 26 2018

nonn

A rooted partition of n is an integer partition of n - 1.

			The a(11) = 8 rooted twice-partitions: (9), (333), (111111111), (4)(4), (22)(22), (1111)(1111), (1)(1)(1)(1)(1), ()()()()()()()()()().

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A002865, A007425, A063834, A093637, A127524, A295931, A300383, A301422, A301462, A301467, A301480, A301706.

Table[If[n===1,1,DivisorSum[n-1,If[#===1,1,DivisorSigma[0,#-1]]&]],{n,100}]

a(n)=if(n==1, 1, sumdiv(n-1, d, if(d==1, 1, numdiv(d-1)))) \\ Andrew Howroyd, Aug 26 2018

cp's OEIS Frontend

A321471 Heinz numbers of integer partitions that can be partitioned into blocks with sums {1, 2, ..., k} for some k.

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A381991 Numbers whose prime indices have a unique multiset partition into constant multisets with distinct sums.

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A321472 Heinz numbers of integer partitions whose parts can be further partitioned and flattened to obtain the partition (k, ..., 3, 2, 1) for some k.

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A321514 Number of ways to choose a factorization of each integer from 2 to n into factors > 1.

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A381637 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into blocks with distinct sums.

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A383014 Numbers whose prime indices can be partitioned into constant blocks with a common sum.

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A301595 Number of thrice-partitions of n.

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A316092 Heinz numbers of integer partitions of prime numbers into prime parts.

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A317143 In the ranked poset of integer partitions ordered by refinement, row n lists the Heinz numbers of integer partitions finer (less) than or equal to the integer partition with Heinz number n.

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