A325368 -id:A325368 - cp's OEIS frontend

A383531 Heinz numbers of integer partitions that do not have distinct multiplicities (Wilf) or distinct nonzero 0-appended differences (conjugate Wilf).

6, 21, 30, 36, 42, 60, 65, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 132, 133, 138, 140, 150, 154, 156, 165, 168, 174, 180, 186, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246, 252, 258, 264, 270, 273, 276, 280, 282, 286, 294, 300, 306, 308

Offset: 1

Gus Wiseman, May 15 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.

An integer partition is Wilf iff its multiplicities are all different (ranked by A130091). It is conjugate Wilf iff its nonzero 0-appended differences are all different (ranked by A383512).

			The terms together with their prime indices begin:
    6: {1,2}
   21: {2,4}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   60: {1,1,2,3}
   65: {3,6}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
  102: {1,2,7}
  105: {2,3,4}
  110: {1,3,5}
  114: {1,2,8}
  120: {1,1,1,2,3}

These partitions are counted by A383530.
Negating both sides gives A383532, counted by A383507.
A048767 is the Look-and-Say transform, union A351294, complement A351295.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions, ranks A130091, conjugate A383512.
A122111 represents conjugation in terms of Heinz numbers.
A325324 counts integer partitions with distinct 0-appended differences, ranks A325367.
A336866 counts non Wilf partitions, ranks A130092, conjugate A383513.
A383709 counts Wilf partitions with distinct 0-appended differences, ranks A383712.
Cf. A001223, A047966, A181819, A238745, A320348, A325325, A325349, A325366, A325368, A325388, A381431, A383506.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],!UnsameQ@@Length/@Split[prix[#]] && !UnsameQ@@Length/@Split[conj[prix[#]]]&]

Equals A130092 /\ A383513.

A384006 Heinz numbers of Look-and-Say partitions without distinct multiplicities (non Wilf).

Original entry on oeis.org

216, 1000, 1296, 2744, 3375, 7776, 9261, 10000, 10648, 17576, 32400, 35937, 38416, 38880, 39304, 42875, 46656, 50625, 54000, 54432, 54872, 59319, 63504, 81000, 85536, 90000, 97336, 100000

Offset: 1

Gus Wiseman, May 19 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.
An integer partition is Wilf iff its multiplicities are all different, ranked by A130091, complement A130092.
An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

			The terms together with their prime indices begin:
     216: {1,1,1,2,2,2}
    1000: {1,1,1,3,3,3}
    1296: {1,1,1,1,2,2,2,2}
    2744: {1,1,1,4,4,4}
    3375: {2,2,2,3,3,3}
    7776: {1,1,1,1,1,2,2,2,2,2}
    9261: {2,2,2,4,4,4}
   10000: {1,1,1,1,3,3,3,3}
   10648: {1,1,1,5,5,5}
   17576: {1,1,1,6,6,6}
   32400: {1,1,1,1,2,2,2,2,3,3}
   35937: {2,2,2,5,5,5}
   38416: {1,1,1,1,4,4,4,4}
   38880: {1,1,1,1,1,2,2,2,2,2,3}
   39304: {1,1,1,7,7,7}
   42875: {3,3,3,4,4,4}
   46656: {1,1,1,1,1,1,2,2,2,2,2,2}
   50625: {2,2,2,2,3,3,3,3}
   54000: {1,1,1,1,2,2,2,3,3,3}
   54432: {1,1,1,1,1,2,2,2,2,2,4}
   54872: {1,1,1,8,8,8}
   59319: {2,2,2,6,6,6}
   63504: {1,1,1,1,2,2,2,2,4,4}
   81000: {1,1,1,2,2,2,2,3,3,3}
   85536: {1,1,1,1,1,2,2,2,2,2,5}
   90000: {1,1,1,1,2,2,3,3,3,3}
   97336: {1,1,1,9,9,9}
  100000: {1,1,1,1,1,3,3,3,3,3}

Ranking sequences are shown in parentheses below.
These partitions are counted by A351592.
For section-sum instead of Look-and-Say we have (A383514), counted by A383506.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).
Cf. A000720, A001223, A048767, A051903, A212166, A325368, A383520, A383531.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],disjointFamilies[prix[#]]!={}&&!UnsameQ@@Last/@FactorInteger[#]&]

A355533 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime(k), then row n is just (k).

Original entry on oeis.org

1, 2, 0, 3, 1, 4, 0, 0, 0, 2, 5, 0, 1, 6, 3, 1, 0, 0, 0, 7, 1, 0, 8, 0, 2, 2, 4, 9, 0, 0, 1, 0, 5, 0, 0, 0, 3, 10, 1, 1, 11, 0, 0, 0, 0, 3, 6, 1, 0, 1, 0, 12, 7, 4, 0, 0, 2, 13, 1, 2, 14, 0, 4, 0, 1, 8, 15, 0, 0, 0, 1, 0, 2, 0

Offset: 2

Gus Wiseman, Jul 12 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.
The version where zero is prepended to the prime indices before taking differences is A287352.
One could argue that row n = 1 is empty, but adding it changes only the offset, with no effect on the data.

			Triangle begins (showing n, prime indices, differences*):
   2:    (1)       1
   3:    (2)       2
   4:   (1,1)      0
   5:    (3)       3
   6:   (1,2)      1
   7:    (4)       4
   8:  (1,1,1)    0 0
   9:   (2,2)      0
  10:   (1,3)      2
  11:    (5)       5
  12:  (1,1,2)    0 1
  13:    (6)       6
  14:   (1,4)      3
  15:   (2,3)      1
  16: (1,1,1,1)  0 0 0
For example, the prime indices of 24 are (1,1,1,2), with differences (0,0,1).

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Crossrefs found in the link are not repeated here.
Row sums are A243056.
The version for prime indices prepended by 0 is A287352.
Constant rows have indices A325328.
Strict rows have indices A325368.
Number of distinct terms in each row are 1 if prime, otherwise A355523.
Row minima are A355525, augmented A355531.
Row maxima are A355526, augmented A355535.
The augmented version is A355534, Heinz number A325351.
The version with prime-indexed rows empty is A355536, Heinz number A325352.
A112798 lists prime indices, sum A056239.
Cf. A001222, A066312, A124010, A286469, A286470, A325160, A325390, A355524.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[PrimeQ[n],{PrimePi[n]},Differences[primeMS[n]]],{n,2,30}]

Row lengths are 1 or A001222(n) - 1 depending on whether n is prime.

A383515 Heinz numbers of integer partitions that are both Look-and-Say and section-sum.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 49, 50, 52, 53, 56, 59, 61, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121, 124, 125

Offset: 1

Gus Wiseman, May 18 2025

nonn

First differs from A383532 in having 325.
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.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.
An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

			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}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  28: {1,1,4}
  29: {10}
  31: {11}
  32: {1,1,1,1,1}

Ranking sequences are shown in parentheses below.
These partitions are counted by A383508.
A048767 is the Look-and-Say transform.
A048768 gives Look-and-Say fixed points, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A381431 is the section-sum transform.
A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).
Cf. A000720, A001223, A047966, A051903, A109298, A212166, A238745, A325368, A351592, A384006.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],disjointFamilies[prix[#]]!={}&&disjointFamilies[conj[prix[#]]]!={}&]

A325457 Heinz numbers of integer partitions with strictly decreasing differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 46, 47, 49, 50, 51, 53, 55, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 75, 77, 79, 82, 83, 85, 86, 87, 89, 91, 93, 94, 95, 97, 98

Offset: 1

Gus Wiseman, May 03 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The enumeration of these partitions by sum is given by A320470.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A056239, A112798, A320470, A320510, A325328, A325352, A325360, A325361, A325368, A325399, A325456, A325461, A320470, A325396.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Greater@@Differences[primeptn[#]]&]

A383516 Heinz numbers of Look-and-Say partitions that are not section-sum partitions.

Original entry on oeis.org

12, 18, 24, 48, 54, 63, 72, 96, 108, 144, 147, 162, 189, 192, 216, 288, 324, 360, 384, 432, 486, 504, 540, 567, 576, 600, 648, 720, 756, 768, 792, 864, 936, 972, 1008, 1029, 1152, 1176, 1188, 1200, 1224, 1296, 1323, 1350, 1368, 1400, 1404, 1440, 1458, 1500

Offset: 1

Gus Wiseman, May 18 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.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.
An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.

			The terms together with their prime indices begin:
   12: {1,1,2}
   18: {1,2,2}
   24: {1,1,1,2}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   63: {2,2,4}
   72: {1,1,1,2,2}
   96: {1,1,1,1,1,2}
  108: {1,1,2,2,2}
  144: {1,1,1,1,2,2}
  147: {2,4,4}
  162: {1,2,2,2,2}
  189: {2,2,2,4}
  192: {1,1,1,1,1,1,2}
  216: {1,1,1,2,2,2}
  288: {1,1,1,1,1,2,2}
  324: {1,1,2,2,2,2}
  360: {1,1,1,2,2,3}
  384: {1,1,1,1,1,1,1,2}

Ranking sequences are shown in parentheses below.
These partitions are counted by A383509.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A381431 is the section-sum transform.
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).
Cf. A000720, A001223, A212166, A238745, A325368, A383514, A383520, A383531, A384006.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],disjointFamilies[prix[#]]!={}&&disjointFamilies[conj[prix[#]]]=={}&]

A383517 Heinz numbers of integer partitions that are neither Look-and-Say nor section-sum partitions.

Original entry on oeis.org

6, 21, 30, 36, 42, 60, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 132, 138, 140, 150, 154, 156, 165, 168, 174, 180, 186, 198, 204, 210, 220, 222, 228, 231, 234, 238, 240, 246, 252, 258, 264, 270, 273, 276, 280, 282, 286, 294, 300, 306, 308, 312, 315

Offset: 1

Gus Wiseman, May 18 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.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432, complement A381433.
An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294, complement A351295.

			The terms together with their prime indices begin:
    6: {1,2}
   21: {2,4}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
  102: {1,2,7}
  105: {2,3,4}
  110: {1,3,5}
  114: {1,2,8}
  120: {1,1,1,2,3}

Ranking sequences are shown in parentheses below.
These partitions are counted by A383510.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A381431 is the section-sum transform.
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383509 counts partitions that are Look-and-Say but not section-sum (A383516).
A383509 counts partitions that are not Look-and-Say but are section-sum (A384007).
Cf. A000720, A001223, A051903, A212166, A238745, A325368, A383514, A383518, A383520, A384006.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],disjointFamilies[prix[#]]=={}&&disjointFamilies[conj[prix[#]]]=={}&]

A383520 Heinz numbers of section-sum partitions with distinct multiplicities (Wilf).

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 20, 23, 25, 27, 28, 29, 31, 32, 37, 40, 41, 43, 44, 45, 47, 49, 50, 52, 53, 56, 59, 61, 64, 67, 68, 71, 73, 75, 76, 79, 80, 81, 83, 88, 89, 92, 97, 98, 99, 101, 103, 104, 107, 109, 112, 113, 116, 117, 121, 124, 125

Offset: 1

Gus Wiseman, May 19 2025

nonn

First differs from A383515 in having 325.
First differs from A383532 in having 325.
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.
An integer partition is Wilf iff its multiplicities are all different, ranked by A130091.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

			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}
   13: {6}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   28: {1,1,4}
   29: {10}
   31: {11}
   32: {1,1,1,1,1}

Ranking sequences are shown in parentheses below.
For non Wilf instead of Wilf we have (A383514), counted by A383506.
These partitions are counted by A383519.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A336866 counts non Wilf partitions (A130092), conjugate (A383513).
A351592 counts non Wilf Look-and-Say partitions, ranked by (A384006).
A381431 is the section-sum transform.
Cf. A000720, A001223, A048767, A051903, A212166, A320348, A325366, A325368.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],disjointFamilies[conj[prix[#]]]!={}&&UnsameQ@@Last/@FactorInteger[#]&]

A384007 Heinz numbers of non Look-and-Say section-sum partitions.

Original entry on oeis.org

10, 14, 15, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 106, 111, 115, 118, 119, 122, 123, 129, 130, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 170, 177, 178, 182, 183, 185, 187, 190

Offset: 1

Gus Wiseman, May 19 2025

nonn

First differs from A383514 in lacking 1000.
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.
An integer partition is Look-and-Say iff it is possible to choose a disjoint family of strict partitions, one of each of its multiplicities. These are ranked by A351294.
An integer partition is section-sum iff it is possible to choose a disjoint family of strict partitions, one of each of its positive 0-appended differences. These are ranked by A381432.

			The terms together with their prime indices begin:
    10: {1,3}    57: {2,8}      94: {1,15}
    14: {1,4}    58: {1,10}     95: {3,8}
    15: {2,3}    62: {1,11}    100: {1,1,3,3}
    22: {1,5}    65: {3,6}     106: {1,16}
    26: {1,6}    69: {2,9}     111: {2,12}
    33: {2,5}    74: {1,12}    115: {3,9}
    34: {1,7}    77: {4,5}     118: {1,17}
    35: {3,4}    82: {1,13}    119: {4,7}
    38: {1,8}    85: {3,7}     122: {1,18}
    39: {2,6}    86: {1,14}    123: {2,13}
    46: {1,9}    87: {2,10}    129: {2,14}
    51: {2,7}    91: {4,6}     130: {1,3,6}
    55: {3,5}    93: {2,11}    133: {4,8}

Ranking sequences are shown in parentheses below.
These partitions are counted by A383509.
Negating both properties gives (A383516).
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions (A351294), complement A351293 (A351295).
A239455 counts section-sum partitions (A381432), complement A351293 (A381433).
A383508 counts partitions that are both Look-and-Say and section-sum (A383515).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
A383511 counts partitions that are Look-and-Say and section-sum but not Wilf (A383518).
Cf. A000720, A001223, A212166, A238745, A325368, A383514, A383520, A383531, A384006.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions/@Length/@Split[y]],UnsameQ@@Join@@#&];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[100],disjointFamilies[prix[#]]=={}&&disjointFamilies[conj[prix[#]]]!={}&]

A325456 Heinz numbers of integer partitions with strictly increasing differences.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87

Offset: 1

Gus Wiseman, May 03 2019

nonn

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
The differences of a sequence are defined as if the sequence were increasing, so for example the differences of (6,3,1) are (-3,-2).
The enumeration of these partitions by sum is given by A240027.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   6: {1,2}
   7: {4}
   9: {2,2}
  10: {1,3}
  11: {5}
  12: {1,1,2}
  13: {6}
  14: {1,4}
  15: {2,3}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Cf. A056239, A112798, A179269, A240026, A240027, A325328, A325352, A325360, A325361, A325368, A325395, A325398, A325457, A325460.

primeptn[n_]:=If[n==1,{},Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],Less@@Differences[primeptn[#]]&]

cp's OEIS Frontend

A383531 Heinz numbers of integer partitions that do not have distinct multiplicities (Wilf) or distinct nonzero 0-appended differences (conjugate Wilf).

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A384006 Heinz numbers of Look-and-Say partitions without distinct multiplicities (non Wilf).

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A355533 Irregular triangle read by rows where row n lists the differences between adjacent prime indices of n; if n is prime(k), then row n is just (k).

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A383515 Heinz numbers of integer partitions that are both Look-and-Say and section-sum.

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A325457 Heinz numbers of integer partitions with strictly decreasing differences.

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A383516 Heinz numbers of Look-and-Say partitions that are not section-sum partitions.

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A383517 Heinz numbers of integer partitions that are neither Look-and-Say nor section-sum partitions.

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A383520 Heinz numbers of section-sum partitions with distinct multiplicities (Wilf).

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A384007 Heinz numbers of non Look-and-Say section-sum partitions.

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