A296150 -id:A296150 - cp's OEIS frontend

A334433 Heinz numbers of all integer partitions sorted first by sum, then by length, and finally lexicographically.

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 15, 14, 18, 20, 24, 32, 13, 25, 21, 22, 27, 30, 28, 36, 40, 48, 64, 17, 35, 33, 26, 45, 50, 42, 44, 54, 60, 56, 72, 80, 96, 128, 19, 49, 55, 39, 34, 75, 63, 70, 66, 52, 81, 90, 100, 84, 88, 108, 120, 112, 144, 160, 192, 256

Offset: 0

Gus Wiseman, Apr 30 2020

First differs from A334435 at a(75) = 99, A334435(75) = 98.
A permutation of the positive integers.
This is the Abramowitz-Stegun ordering of integer partitions when the parts are read in the usual (weakly decreasing) order. The case of reversed (weakly increasing) partitions is A185974.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}            32: {1,1,1,1,1}       42: {1,2,4}
    2: {1}           13: {6}               44: {1,1,5}
    3: {2}           25: {3,3}             54: {1,2,2,2}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           22: {1,5}             56: {1,1,1,4}
    6: {1,2}         27: {2,2,2}           72: {1,1,1,2,2}
    8: {1,1,1}       30: {1,2,3}           80: {1,1,1,1,3}
    7: {4}           28: {1,1,4}           96: {1,1,1,1,1,2}
    9: {2,2}         36: {1,1,2,2}        128: {1,1,1,1,1,1,1}
   10: {1,3}         40: {1,1,1,3}         19: {8}
   12: {1,1,2}       48: {1,1,1,1,2}       49: {4,4}
   16: {1,1,1,1}     64: {1,1,1,1,1,1}     55: {3,5}
   11: {5}           17: {7}               39: {2,6}
   15: {2,3}         35: {3,4}             34: {1,7}
   14: {1,4}         33: {2,5}             75: {2,3,3}
   18: {1,2,2}       26: {1,6}             63: {2,2,4}
   20: {1,1,3}       45: {2,2,3}           70: {1,3,4}
   24: {1,1,1,2}     50: {1,3,3}           66: {1,2,5}
Triangle begins:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  22  27  30  28  36  40  48  64
  17  35  33  26  45  50  42  44  54  60  56  72  80  96 128
This corresponds to the tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(21)(111)
        (4)(22)(31)(211)(1111)
  (5)(32)(41)(221)(311)(2111)(11111)

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
Compositions under the same order are A124734 (triangle).
The version for reversed (weakly increasing) partitions is A185974.
The constructive version is A334301.
Ignoring length gives A334434, or A334437 for reversed partitions.
The dual version (sum/length/revlex) is A334438.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colexicographic order (sum/length/colex) are A036037.
Graded reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded lexicographically ordered partitions are A193073.
Graded Heinz numbers are A215366.
Sorting partitions by Heinz number gives A296150.
Partitions in increasing-length reverse-lexicographic order (sum/length/revlex) are A334439 (not A036037).
Cf. A026791, A036043, A048793, A056239, A129129, A211992, A228351, A228531, A334302, A334435, A334436.

Join@@Table[Times@@Prime/@#&/@Sort[IntegerPartitions[n]],{n,0,8}]

A001222(a(n)) = A036043(n).

A357982 Replace prime(k) with A000009(k) in the prime factorization of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 2, 3, 8, 1, 4, 4, 1, 2, 10, 2, 12, 1, 3, 5, 4, 1, 15, 6, 4, 2, 18, 2, 22, 3, 2, 8, 27, 1, 4, 4, 5, 4, 32, 1, 6, 2, 6, 10, 38, 2, 46, 12, 2, 1, 8, 3, 54, 5, 8, 4, 64, 1, 76, 15, 4, 6, 6, 4, 89, 2, 1

Offset: 1

Gus Wiseman, Oct 25 2022

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. This sequence gives the number of ways to choose a strict partition of each prime index of n.

The indices i, where a(i) = 1, form A003586, and the indices j, where a(j) > 1, form A059485. - Ivan N. Ianakiev, Oct 27 2022

			The a(121) = 9 twice-partitions are: (5)(5), (5)(41), (5)(32), (41)(5), (41)(41), (41)(32), (32)(5), (32)(41), (32)(32).

Other multiplicative sequences: A003961, A357852, A064988, A064989, A357980.
The non-strict version is A299200.
A horizontal version is A357978, non-strict A357977.
A000040 lists the primes.
A056239 adds up prime indices, row-sums of A112798.
Cf. A000041, A000720, A003964, A063834, A076610, A215366, A273873, A296150, A299201-A299203, A357975, A357979, A357983.
Cf. A003586, A059485.

Table[Times@@Cases[FactorInteger[n],{p_,k_}:>PartitionsQ[PrimePi[p]]^k],{n,100}]

f9(n) = polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)), n); \\ A000009
a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = f9(primepi(f[k,1]))); factorback(f); \\ Michel Marcus, Oct 26 2022

A129129 An irregular triangular array of natural numbers read by rows, with shape sequence A000041(n) related to sequence A060850.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 10, 9, 12, 16, 11, 14, 15, 20, 18, 24, 32, 13, 22, 21, 28, 25, 30, 40, 27, 36, 48, 64, 17, 26, 33, 44, 35, 42, 56, 50, 45, 60, 80, 54, 72, 96, 128, 19, 34, 39, 52, 55, 66, 88, 49, 70, 63, 84, 112, 75, 100, 90, 120, 160, 81, 108, 144, 192, 256

Offset: 0

Alford Arnold, Mar 31 2007

The tree begins (at height n, n >= 0, nodes represent partitions of n)
0:  1
1:  2
2:  3  4
3:  5  6  8
4:  7 10  9 12 16
5: 11 14 15 20 18 24 32
...
and hence differs from A114622.
Ordering [graded reverse lexicographic order] of partitions (positive integer representation) of nonnegative integers, where part of size i [as summand] is mapped to i-th prime [as multiplicand], where the empty partition for 0 yields the empty product, i.e., 1. Permutation of positive integers, since bijection [1-1 and onto map] between the set of all partitions of nonnegative integers and positive integers. - Daniel Forgues, Aug 07 2018
These are all Heinz numbers of integer partitions in graded reverse-lexicographic order, where The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This is the so-called "Mathematica" order (sum/revlex) of partitions (A080577). Partitions in lexicographic order (sum/lex) are A193073, with Heinz numbers A334434. - Gus Wiseman, May 19 2020

			The array is a tree structure as described by A128628. If a node value has only one branch the value is twice that of its parent node. If it has two branches one is twice that of its parent node but the other is defined as indicated below:
(1) pick an odd number (e.g., 135)
(2) calculate its prime factorization (135 = 5*3*3*3)
(3) note the least prime factor (LPF(135) = 3)
(4) note the index of the LPF (index(3) = 2)
(5) subtract one from the index (2-1 = 1)
(6) calculate the prime associated with the value in step five (prime(1) = 2)
(7) The parent node of the odd number 135 is (2/3)*135 = 90 = A252461(135).
From _Daniel Forgues_, Aug 07 2018: (Start)
Partitions of 4 in graded reverse lexicographic order:
{4}: p_4 = 7;
{3,1}: p_3 * p_1 = 5 * 2 = 10;
{2,2}: p_2 * p_2 = 3^2 = 9;
{2,1,1}: p_2 * p_1 * p_1 = 3 * 2^2 = 12;
{1,1,1,1}: p_1 * p_1 * p_1 * p_1 = 2^4 = 16. (End)
From _Gus Wiseman_, May 19 2020: (Start)
The sequence together with the corresponding partitions begins:
    1: ()            24: (2,1,1,1)         35: (4,3)
    2: (1)           32: (1,1,1,1,1)       42: (4,2,1)
    3: (2)           13: (6)               56: (4,1,1,1)
    4: (1,1)         22: (5,1)             50: (3,3,1)
    5: (3)           21: (4,2)             45: (3,2,2)
    6: (2,1)         28: (4,1,1)           60: (3,2,1,1)
    8: (1,1,1)       25: (3,3)             80: (3,1,1,1,1)
    7: (4)           30: (3,2,1)           54: (2,2,2,1)
   10: (3,1)         40: (3,1,1,1)         72: (2,2,1,1,1)
    9: (2,2)         27: (2,2,2)           96: (2,1,1,1,1,1)
   12: (2,1,1)       36: (2,2,1,1)        128: (1,1,1,1,1,1,1)
   16: (1,1,1,1)     48: (2,1,1,1,1)       19: (8)
   11: (5)           64: (1,1,1,1,1,1)     34: (7,1)
   14: (4,1)         17: (7)               39: (6,2)
   15: (3,2)         26: (6,1)             52: (6,1,1)
   20: (3,1,1)       33: (5,2)             55: (5,3)
   18: (2,2,1)       44: (5,1,1)           66: (5,2,1)
(End)

Michael De Vlieger, Table of n, a(n) for n = 0..11731 (rows 0 <= n <= 26).
OEIS Wiki, Partitions#Orderings of partitions.
Wikiversity, Lexicographic and colexicographic order.

Cf. A080577 (the partitions), A252461, A114622, A128628, A215366 (sorted rows).
Row lengths are A000041.
Compositions under the same order are A066099.
The opposite version (sum/lex) is A334434.
The length-sensitive version (sum/length/revlex) is A334438.
The version for reversed (weakly increasing) partitions is A334436.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun order (sum/length/lex) are A036036.
Sum of prime indices is A056239.
Sorting reversed partitions by Heinz number gives A112798.
Partitions in lexicographic order are A193073.
Sorting partitions by Heinz number gives A296150.
Cf. A036037, A185974, A211992, A334301, A334433, A334435, A334437, A334439.

b:= (n, i)-> `if`(n=0 or i=1, [2^n], [map(x-> x*ithprime(i),
                b(n-i, min(n-i, i)))[], b(n, i-1)[]]):
T:= n-> b(n$2)[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Feb 14 2020

Array[Times @@ # & /@ Prime@ IntegerPartitions@ # &, 9, 0] // Flatten (* Michael De Vlieger, Aug 07 2018 *)
b[n_, i_] := b[n, i] = If[n == 0 || i == 1, {2^n}, Join[(# Prime[i]&) /@ b[n - i, Min[n - i, i]], b[n, i - 1]]];
T[n_] := b[n, n];
T /@ Range[0, 10] // Flatten (* Jean-François Alcover, May 21 2021, after Alois P. Heinz *)

From Gus Wiseman, May 19 2020: (Start)
A001222(a(n)) = A238966(n).
A001221(a(n)) = A115623(n).
A056239(a(n)) = A036042(n).
A061395(a(n)) = A331581(n).
(End)

A304793 Number of distinct positive subset-sums of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 4, 1, 3, 3, 4, 1, 5, 1, 5, 3, 3, 1, 5, 2, 3, 3, 5, 1, 6, 1, 5, 3, 3, 3, 6, 1, 3, 3, 6, 1, 7, 1, 5, 5, 3, 1, 6, 2, 5, 3, 5, 1, 7, 3, 7, 3, 3, 1, 7, 1, 3, 4, 6, 3, 7, 1, 5, 3, 6, 1, 7, 1, 3, 5, 5, 3, 7, 1, 7, 4, 3, 1, 8, 3, 3, 3, 7, 1, 8, 3, 5, 3, 3, 3, 7, 1, 5, 5, 8, 1, 7, 1, 7, 7

Offset: 1

Gus Wiseman, May 18 2018

nonn

A positive integer n is a positive subset-sum of an integer partition y if there exists a submultiset of y with sum n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
a(n) <= A000005(n).
One less than the number of distinct values obtained when A056239 is applied to all divisors of n. - Antti Karttunen, Jul 01 2018

			The positive subset-sums of (4,3,1) are {1, 3, 4, 5, 7, 8} so a(70) = 6.
The positive subset-sums of (5,1,1,1) are {1, 2, 3, 5, 6, 7, 8} so a(88) = 7.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A056239, A122768, A276024, A284640, A296150, A299701, A299702, A301855, A301935, A301957, A304792, A304795, A305611.

Table[Length[Union[Total/@Rest[Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]]],{n,100}]

up_to = 65537;
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
v056239 = vector(up_to,n,A056239(n));
A304793(n) = { my(m=Map(),s,k=0); fordiv(n,d,if(!mapisdefined(m,s = v056239[d]), mapput(m,s,s); k++)); (k-1); }; \\ Antti Karttunen, Jul 01 2018

More terms from Antti Karttunen, Jul 01 2018

A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

1, 2, 1, 1, 3, 1, 2, 1, 1, 1, 4, 2, 2, 1, 3, 1, 1, 2, 1, 1, 1, 1, 5, 2, 3, 1, 4, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 6, 3, 3, 2, 4, 1, 5, 2, 2, 2, 1, 2, 3, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 7, 3, 4, 2, 5, 1, 6, 2, 2, 3

Offset: 0

Gus Wiseman, Apr 30 2020

			The sequence of all reversed partitions begins:
  ()         (1,4)        (1,1,1,1,2)
  (1)        (1,2,2)      (1,1,1,1,1,1)
  (2)        (1,1,3)      (7)
  (1,1)      (1,1,1,2)    (3,4)
  (3)        (1,1,1,1,1)  (2,5)
  (1,2)      (6)          (1,6)
  (1,1,1)    (3,3)        (2,2,3)
  (4)        (2,4)        (1,3,3)
  (2,2)      (1,5)        (1,2,4)
  (1,3)      (2,2,2)      (1,1,5)
  (1,1,2)    (1,2,3)      (1,2,2,2)
  (1,1,1,1)  (1,1,4)      (1,1,2,3)
  (5)        (1,1,2,2)    (1,1,1,4)
  (2,3)      (1,1,1,3)    (1,1,1,2,2)
This sequence can also be interpreted as the following triangle, whose n-th row is itself a finite triangle with A000041(n) rows.
                            0
                           (1)
                        (2) (1,1)
                    (3) (1,2) (1,1,1)
            (4) (2,2) (1,3) (1,1,2) (1,1,1,1)
  (5) (2,3) (1,4) (1,2,2) (1,1,3) (1,1,1,2) (1,1,1,1,1)
Showing partitions as their Heinz numbers (see A334435) gives:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  22  27  30  28  36  40  48  64
  17  35  33  26  45  50  42  44  54  60  56  72  80  96 128

OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

Row lengths are A036043.
Lexicographically ordered reversed partitions are A026791.
The dual ordering (sum/length/lex) of reversed partitions is A036036.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Lexicographically ordered partitions are A193073.
Graded Heinz numbers are A215366.
Ignoring length gives A228531.
Sorting partitions by Heinz number gives A296150.
The version for compositions is A296774.
The dual ordering (sum/length/lex) of non-reversed partitions is A334301.
Taking Heinz numbers gives A334435.
The version for regular (non-reversed) partitions is A334439 (not A036037).
Cf. A000041, A048793, A066099, A080576, A124734, A162247, A211992, A228100, A228351.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

A299729 Heinz numbers of non-knapsack partitions.

Original entry on oeis.org

12, 24, 30, 36, 40, 48, 60, 63, 70, 72, 80, 84, 90, 96, 108, 112, 120, 126, 132, 140, 144, 150, 154, 156, 160, 165, 168, 180, 189, 192, 198, 200, 204, 210, 216, 220, 224, 228, 240, 252, 264, 270, 273, 276, 280, 286, 288, 300, 308, 312, 315, 320, 324, 325

Offset: 1

Gus Wiseman, Feb 17 2018

nonn

An integer partition is non-knapsack if there exist two different submultisets with the same sum. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			12 is the Heinz number of (2,1,1) which is not knapsack because 2 = 1 + 1.

Cf. A056239, A108917, A112798, A275972, A276024, A284640, A296150, A299701, A299702.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],!UnsameQ@@Plus@@@Union[Rest@Subsets[primeMS[#]]]&]

A334435 Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 7, 9, 10, 12, 16, 11, 15, 14, 18, 20, 24, 32, 13, 25, 21, 22, 27, 30, 28, 36, 40, 48, 64, 17, 35, 33, 26, 45, 50, 42, 44, 54, 60, 56, 72, 80, 96, 128, 19, 49, 55, 39, 34, 75, 63, 70, 66, 52, 81, 90, 100, 84, 88, 108, 120, 112, 144, 160, 192, 256

Offset: 0

Gus Wiseman, May 02 2020

First differs from A334433 at a(75) = 99, A334433(75) = 98.
First differs from A334436 at a(22) = 22, A334436(22) = 27.
A permutation of the positive integers.
Reversed integer partitions are finite weakly increasing sequences of positive integers.
This is the Abramowitz-Stegun ordering of reversed partitions (A185974) except that the finer order is reverse-lexicographic instead of lexicographic. The version for non-reversed partitions is A334438.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
As a triangle with row lengths A000041, the sequence starts {{1},{2},{3,4},{5,6,8},...}, so offset is 0.

			The sequence of terms together with their prime indices begins:
    1: {}            32: {1,1,1,1,1}       42: {1,2,4}
    2: {1}           13: {6}               44: {1,1,5}
    3: {2}           25: {3,3}             54: {1,2,2,2}
    4: {1,1}         21: {2,4}             60: {1,1,2,3}
    5: {3}           22: {1,5}             56: {1,1,1,4}
    6: {1,2}         27: {2,2,2}           72: {1,1,1,2,2}
    8: {1,1,1}       30: {1,2,3}           80: {1,1,1,1,3}
    7: {4}           28: {1,1,4}           96: {1,1,1,1,1,2}
    9: {2,2}         36: {1,1,2,2}        128: {1,1,1,1,1,1,1}
   10: {1,3}         40: {1,1,1,3}         19: {8}
   12: {1,1,2}       48: {1,1,1,1,2}       49: {4,4}
   16: {1,1,1,1}     64: {1,1,1,1,1,1}     55: {3,5}
   11: {5}           17: {7}               39: {2,6}
   15: {2,3}         35: {3,4}             34: {1,7}
   14: {1,4}         33: {2,5}             75: {2,3,3}
   18: {1,2,2}       26: {1,6}             63: {2,2,4}
   20: {1,1,3}       45: {2,2,3}           70: {1,3,4}
   24: {1,1,1,2}     50: {1,3,3}           66: {1,2,5}
Triangle begins:
   1
   2
   3   4
   5   6   8
   7   9  10  12  16
  11  15  14  18  20  24  32
  13  25  21  22  27  30  28  36  40  48  64
  17  35  33  26  45  50  42  44  54  60  56  72  80  96 128
This corresponds to the following tetrangle:
                  0
                 (1)
               (2)(11)
             (3)(12)(111)
        (4)(22)(13)(112)(1111)
  (5)(23)(14)(122)(113)(1112)(11111)

Wikiversity, Lexicographic and colexicographic order

Row lengths are A000041.
The dual version (sum/length/lex) is A185974.
Compositions under the same order are A296774 (triangle).
The constructive version is A334302.
Ignoring length gives A334436.
The version for non-reversed partitions is A334438.
Partitions in this order (sum/length/revlex) are A334439.
Lexicographically ordered reversed partitions are A026791.
Reversed partitions in Abramowitz-Stegun (sum/length/lex) order are A036036.
Partitions in increasing-length colex order (sum/length/colex) are A036037.
Reverse-lexicographically ordered partitions are A080577.
Sorting reversed partitions by Heinz number gives A112798.
Graded lexicographically ordered partitions are A193073.
Partitions in colexicographic (sum/colex) order are A211992.
Graded Heinz numbers are given by A215366.
Sorting partitions by Heinz number gives A296150.
Cf. A056239, A124734, A129129, A228100, A228531, A333219, A333220, A334301, A334433, A334434, A334437.

revlensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]

A001222(a(n)) = A036043(n).

A353840 Trajectory of the partition run-sum transformation of n, using Heinz numbers.

Original entry on oeis.org

1, 2, 3, 4, 3, 5, 6, 7, 8, 5, 9, 7, 10, 11, 12, 9, 7, 13, 14, 15, 16, 7, 17, 18, 14, 19, 20, 15, 21, 22, 23, 24, 15, 25, 13, 26, 27, 13, 28, 21, 29, 30, 31, 32, 11, 33, 34, 35, 36, 21, 37, 38, 39, 40, 25, 13, 41, 42, 43, 44, 33, 45, 35, 46, 47, 48, 21, 49, 19

Offset: 1

Gus Wiseman, May 25 2022

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 run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 given in row 12 corresponds to the partitions (2,1,1) -> (2,2) -> (4).
This is the iteration of the transformation f described by Kimberling at A237685.

			Triangle begins:
   1
   2
   3
   4  3
   5
   6
   7
   8  5
   9  7
  10
  11
  12  9  7
Row 87780 is the following trajectory (left column), with prime indices shown on the right:
  87780: {1,1,2,3,4,5,8}
  65835: {2,2,3,4,5,8}
  51205: {3,4,4,5,8}
  19855: {3,5,8,8}
   2915: {3,5,16}

The version for run-lengths instead of sums is A325239 or A325277.
This is the iteration of A353832, with composition version A353847.
Row-lengths are A353841, counted by A353846.
Final terms are A353842.
Counting rows by final omega gives A353843.
Rows ending in a prime number are A353844, counted by A353845.
These sequences for compositions are A353853-A353859.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A124010 gives prime signature, sorted A118914.
A182850 or A323014 gives frequency depth.
A300273 ranks collapsible partitions, counted by A275870.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353862 gives greatest run-sum of prime indices, least A353931.
Cf. A071625, A073093, A181819, A323023, A353743, A353834, A353864, A353865, A353866, A353867.

Table[NestWhileList[Times@@Prime/@Cases[FactorInteger[#],{p_,k_}:>PrimePi[p]*k]&,n,Not@*SquareFreeQ],{n,30}]

A299201 Number of twice-partitions whose composite is the integer partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 5, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 8, 2, 2, 3, 4, 1, 6, 1, 7, 2, 2, 2, 11, 1, 2, 2, 8, 1, 5, 1, 4, 4, 2, 1, 16, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 13, 1, 2, 5, 11, 2, 5, 1, 4, 2, 6, 1, 19, 1, 2, 4, 4, 2, 5, 1, 13, 5, 2, 1, 13, 2

Offset: 1

Gus Wiseman, Feb 05 2018

nonn

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

			The a(36) = 11 twice-partitions:
  (2211),
  (22)(11), (211)(2), (221)(1), (21)(21),
  (2)(2)(11), (2)(11)(2), (11)(2)(2), (22)(1)(1), (21)(2)(1),
  (2)(2)(1)(1).

Robert Price, Table of n, a(n) for n = 1..200

Cf. A000041, A063834, A112798, A196545, A273873, A281145, A289501, A290261, A296150, A299200, A299202, A299203.

nn=100;
ptns=Table[If[n===1,{},Join@@Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]],{n,nn}];
tris=Join@@Map[Tuples[IntegerPartitions/@#]&,ptns];
Table[Length[Select[tris,Sort[Join@@#,Greater]===y&]],{y,ptns}]

A228100 Triangle in which n-th row lists all partitions of n, such that partitions of n into m parts appear in lexicographic order previous to the partitions of n into k parts if k < m. (Fenner-Loizou tree.)

Original entry on oeis.org

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

Offset: 1

Peter Luschny, Aug 10 2013

First differs from A193073 at a(58). - Omar E. Pol, Sep 22 2013

The partition lengths appear to be A331581. - Gus Wiseman, May 12 2020

			The sixth row is:
[1, 1, 1, 1, 1, 1]
[2, 1, 1, 1, 1]
[2, 2, 1, 1]
[3, 1, 1, 1]
[2, 2, 2]
[3, 2, 1]
[4, 1, 1]
[3, 3]
[4, 2]
[5, 1]
[6]
From _Gus Wiseman_, May 10 2020: (Start)
The triangle with partitions shown as Heinz numbers (A333485) begins:
    1
    2
    4   3
    8   6   5
   16  12   9  10   7
   32  24  18  20  15  14  11
   64  48  36  40  27  30  28  25  21  22  13
  128  96  72  80  54  60  56  45  50  42  44  35  33  26  17
(End)

T. I. Fenner, G. Loizou: A binary tree representation and related algorithms for generating integer partitions. The Computer J. 23(4), 332-337 (1980)
D. E. Knuth: The Art of Computer Programming. Generating all combinations and partitions, vol. 4, fasc. 3, 7.2.1.4, exercise 10.
K. Yamanaka, Y. Otachi, Sh. Nakano: Efficient enumeration of ordered trees with k leaves. In: WALCOM: Algorithms and Computation, Lecture Notes in Computer Science Volume 5431, 141-150 (2009)
S. Zaks, D. Richards: Generating trees and other combinatorial objects lexicographically. SIAM J. Comput. 8(1), 73-81 (1979)
A. Zoghbi, I. Stojmenovic': Fast algorithms for generating integer partitions. Int. J. Comput. Math. 70, 319-332 (1998)

Alois P. Heinz, rows n = 1..19, flattened
Peter Luschny, Integer Partition Trees
OEIS Wiki, Orderings of partitions
Wikiversity, Lexicographic and colexicographic order

See A036036 for the Hindenburg (graded reflected colexicographic) ordering.
See A036037 for the graded colexicographic ordering.
See A080576 for the Maple (graded reflected lexicographic) ordering.
See A080577 for the Mathematica (graded reverse lexicographic) ordering.
See A182937 the Fenner-Loizou (binary tree in preorder traversal) ordering.
See A193073 for the graded lexicographic ordering.
The version for compositions is A296773.
Taking Heinz numbers gives A333485.
Lexicographically ordered reversed partitions are A026791.
Sorting partitions by Heinz number gives A296150, or A112798 for reversed partitions.
Reversed partitions under the (sum/length/revlex) ordering are A334302.
Cf. A000041, A036043, A048793, A211992, A228351, A228531, A331581, A333484, A334301, A334439.

b:= proc(n, i) b(n, i):= `if`(n=0 or i=1, [[1$n]], [b(n, i-1)[],
      `if`(i>n, [], map(x-> [i, x[]], b(n-i, i)))[]])
    end:
T:= n-> map(h-> h[], sort(b(n$2), proc(x, y) local i;
        if nops(x)<>nops(y) then return nops(x)>nops(y) else
        for i to nops(x) do if x[i]<>y[i] then return x[i]Alois P. Heinz, Aug 13 2013

row[n_] := Flatten[Reverse[Sort[#]]& /@ SplitBy[Sort[IntegerPartitions[n] ], Length], 1] // Reverse; Array[row, 8] // Flatten (* Jean-François Alcover, Dec 05 2016 *)
ralensort[f_,c_]:=If[Length[f]!=Length[c],Length[f]>Length[c],OrderedQ[{f,c}]];
Join@@Table[Sort[IntegerPartitions[n],ralensort],{n,0,8}] (* Gus Wiseman, May 10 2020 *)

from collections import deque
def GeneratePartitions(n, visit):
    p = ([], 0, n)
    queue = deque()
    queue.append(p)
    visit(p)
    while len(queue) > 0 :
        (phead, pheadLen, pnum1s) = queue.popleft()
        if pnum1s != 1 :
            head = phead[:pheadLen] + [2]
            q = (head, pheadLen + 1, pnum1s - 2)
            if 1 <= q[2] : queue.append(q)
            visit(q)
        if pheadLen == 1 or (pheadLen > 1 and \
                      (phead[pheadLen - 1] != phead[pheadLen - 2])) :
            head = phead[:pheadLen]
            head[pheadLen - 1] += 1
            q = (head, pheadLen, pnum1s - 1)
            if 1 <= q[2] : queue.append(q)
            visit(q)
def visit(q): print(q[0] + [1 for i in range(q[2])])
for n in (1..7): GeneratePartitions(n, visit)

cp's OEIS Frontend

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A304793 Number of distinct positive subset-sums of the integer partition with Heinz number n.

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A334302 Irregular triangle read by rows where row k is the k-th reversed integer partition, if reversed partitions are sorted first by sum, then by length, and finally reverse-lexicographically.

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A299729 Heinz numbers of non-knapsack partitions.

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A334435 Heinz numbers of all reversed integer partitions sorted first by sum, then by length, and finally reverse-lexicographically.

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A353840 Trajectory of the partition run-sum transformation of n, using Heinz numbers.

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