A130092 -id:A130092 - cp's OEIS frontend

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

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[#]]]!={}&]

A383508 Number of integer partitions of n that are both Look-and-Say and section-sum partitions.

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 6, 7, 9, 12, 14, 19, 22, 27, 30, 35, 42, 50, 58, 68, 82, 92, 112, 126, 149, 174, 199, 225, 263, 299, 337, 388, 435, 488, 545, 635, 681, 775, 841, 948, 1051, 1181, 1271, 1446, 1553, 1765, 1896, 2141, 2285, 2608, 2799

Offset: 0

Gus Wiseman, May 17 2025

nonn

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 its conjugate is Look-and-Say, meaning 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 a(1) = 1 through a(8) = 9 partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (311)    (33)      (322)      (44)
                    (1111)  (11111)  (222)     (331)      (332)
                                     (411)     (511)      (611)
                                     (3111)    (4111)     (2222)
                                     (111111)  (31111)    (5111)
                                               (1111111)  (41111)
                                                          (311111)
                                                          (11111111)

Ranking sequences are shown in parentheses below.
The non Wilf case is A383511 (A383518).
These partitions are ranked by (A383515).
A000041 counts integer partitions, strict A000009.
A047993 counts partitions with max part = length (A106529).
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).
A351592 counts non Wilf Look-and-Say partitions (A384006).
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).
Cf. A033461, A047966, A048767, A320348, A325324, A325325, A381431, A383519, A383530, A383709.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions /@ Length/@Split[y]],UnsameQ@@Join@@#&];
conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]!={}&&disjointFamilies[conj[#]]!={}&]], {n,0,30}]

A383510 Number of integer partitions of n that are neither Look-and-Say nor section-sum.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 3, 2, 5, 10, 14, 19, 33, 38, 55, 81, 107, 137, 201, 248, 349, 450, 596, 745, 1000, 1242, 1611, 2007, 2567, 3164, 4025, 4920, 6166, 7545, 9347, 11360, 14004, 16932, 20686, 24949, 30305, 36366, 43939, 52521, 63098, 75221

Offset: 0

Gus Wiseman, May 18 2025

nonn

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 its conjugate is Look-and-Say, meaning 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 a(3) = 1 through a(10) = 14 partitions:
  (21)  .  .  (42)    (421)   (431)    (432)     (532)
              (321)   (3211)  (521)    (531)     (541)
              (2211)          (3221)   (621)     (721)
                              (4211)   (3321)    (4321)
                              (32111)  (4221)    (5221)
                                       (4311)    (5311)
                                       (5211)    (6211)
                                       (32211)   (32221)
                                       (42111)   (33211)
                                       (321111)  (42211)
                                                 (43111)
                                                 (52111)
                                                 (421111)
                                                 (3211111)

Ranking sequences are shown in parentheses below.
These partitions are ranked by (A383517).
A000041 counts integer partitions, strict A000009.
A048767 is the Look-and-Say transform, fixed points A048768, counted by A217605.
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).
A351592 counts non Wilf Look-and-Say partitions (A384006).
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).
A383519 counts section-sum Wilf partitions (A383520).
A383530 counts partitions that are neither Wilf nor conjugate Wilf (A383531).
Cf. A033461, A047966, A098859, A122111, A320348, A325324, A325325, A383511, A383709.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions /@ Length/@Split[y]],UnsameQ@@Join@@#&];
conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]], {k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]=={} && disjointFamilies[conj[#]]=={}&]], {n,0,15}]

A383511 Number of integer partitions of n that are Look-and-Say and section-sum but not Wilf.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 2, 1, 0, 1, 0, 3, 3, 0, 0, 5, 2, 1, 5, 6, 1, 10, 5, 12, 11, 12, 14, 31, 15, 25, 28, 38

Offset: 0

Gus Wiseman, May 18 2025

A 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.
A partition is section-sum iff its conjugate is Look-and-Say, meaning 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.
A partition is Wilf iff its multiplicities are all different (ranked by A130091).

			The a(n) partitions for n = 12, 15, 20, 24, 28:
  (6,3,3)  (6,6,3)    (8,8,4)    (12,6,6)         (14,7,7)
           (6,3,3,3)  (10,5,5)   (6,6,6,3,3)      (8,8,8,4)
                      (8,4,4,4)  (8,4,4,4,4)      (8,8,4,4,4)
                                 (6,6,3,3,3,3)    (8,4,4,4,4,4)
                                 (6,3,3,3,3,3,3)  (10,6,6,2,2,2)
                                                  (11,6,6,1,1,1,1,1)

Ranking sequences are shown in parentheses below.
This is the non Wilf case of A383508 (A383515).
These partitions are ranked by (A383518).
A000041 counts integer partitions, strict A000009.
A098859 counts Wilf partitions (A130091), conjugate (A383512).
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).
A351592 counts non Wilf Look-and-Say partitions (A384006).
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).
A383519 counts section-sum Wilf partitions (A383520).
Cf. A033461, A047966, A048767, A122111, A217605, A320348, A325324, A325325, A381431, A383530, A383709.

disjointFamilies[y_]:=Select[Tuples[IntegerPartitions /@ Length/@Split[y]],UnsameQ@@Join@@#&];
conj[y_]:=If[Length[y]==0,y, Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n], disjointFamilies[#]!={}&&disjointFamilies[conj[#]]!={} && !UnsameQ@@Length/@Split[#]&]], {n,0,30}]

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[#]&]

A380958 Number of prime factors of n (with multiplicity) minus sum of distinct prime exponents of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 1, 2, 0, 1, 1, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 2, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Feb 13 2025

nonn

			The prime factors of 2100 are {2,2,3,5,5,7}, with distinct multiplicities {1,2}, so a(2100) = 6 - (1+2) = 3.

Positions of 0's are A130091, complement A130092.
The RHS (sum of distinct prime exponents) is A136565.
For prime factors instead of exponents see A280292, firsts A280286, sorted A381075.
For prime indices instead of exponents see A380955, firsts A380956, sorted A380957.
Position of first appearance of n is A380989(n).
A000040 lists the primes, differences A001223.
A005117 lists squarefree numbers, complement A013929.
A005361 gives product of prime signature.
A055396 gives least prime index, greatest A061395.
A056239 (reverse A296150) adds up prime indices, row sums of A112798, counted by A001222.
A124010 lists prime exponents (signature); see A001222, A001221, A051903, A051904.
Cf. A000720, A046660, A071625, A075254, A075255, A076694, A081770, A116861, A178503, A290106, A380986.

Table[PrimeOmega[n]-Total[Union[Last/@If[n==1,{},FactorInteger[n]]]],{n,100}]

a(n) = A001222(n) - A136565(n).

A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.

Original entry on oeis.org

325, 845, 931, 1625, 2527, 3509, 6253, 6517, 8125, 9251

Offset: 1

Gus Wiseman, May 18 2025

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

			The terms together with their prime indices begin:
   325: {3,3,6}
   845: {3,6,6}
   931: {4,4,8}
  1625: {3,3,3,6}
  2527: {4,8,8}
  3509: {5,5,10}
  6253: {6,6,12}
  6517: {4,4,4,8}
  8125: {3,3,3,3,6}
  9251: {5,10,10}

Ranking sequences are shown in parentheses below.
These partitions are counted by A383511.
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).
A383510 counts partitions that are neither Look-and-Say nor section-sum (A383517).
Cf. A047966, A051903, A212166, A238745, A325368, A351592, 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}]]]];
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Select[Range[1000],disjointFamilies[prix[#]]!={}&&disjointFamilies[conj[prix[#]]]!={}&&!UnsameQ@@Length/@Split[conj[prix[#]]]&]

A380989 Position of first appearance of n in A380958 (number of prime factors minus sum of distinct prime exponents).

Original entry on oeis.org

1, 6, 30, 210, 900, 7776, 27000, 279936, 810000, 9261000, 24300000, 362797056, 729000000, 13060694016, 21870000000, 408410100000, 656100000000, 16926659444736, 19683000000000, 609359740010496, 590490000000000, 18010885410000000, 17714700000000000

Offset: 0

Gus Wiseman, Feb 18 2025

nonn

Is this sequence strictly increasing?
From David Consiglio, Jr., Feb 20 2025: (Start)
The answer to the question above is: no, a(21) < a(20). And all subsequent odd indexed terms are lower than their even predecessors.
All terms must be a product of x primes (with multiplicity) to the y power where x-y = n and x mod y = 0. There are very few combinations of numbers that meet these criteria, so checking all of them to find the minimum outcome is quite fast.
Example --> n=5
6 primes to the 1 power --> 6 distinct primes
  2*3*5*7*11*13 = 30030
7 primes to the 2 power -- disallowed (5 mod 2 = 1)
8 primes to the 3 power -- disallowed (4 mod 3 = 1)
9 primes to the 4 power -- disallowed (9 mod 4 = 1)
10 primes to the 5 power --> 2 distinct primes
  2*2*2*2*2*3*3*3*3*3 = 7776
The minimum value is 7776 and thus a(5) = 7776. (End)

			The terms together with their prime indices begin:
        1: {}
        6: {1,2}
       30: {1,2,3}
      210: {1,2,3,4}
      900: {1,1,2,2,3,3}
     7776: {1,1,1,1,1,2,2,2,2,2}
    27000: {1,1,1,2,2,2,3,3,3}
   279936: {1,1,1,1,1,1,1,2,2,2,2,2,2,2}
   810000: {1,1,1,1,2,2,2,2,3,3,3,3}
  9261000: {1,1,1,2,2,2,3,3,3,4,4,4}

David Consiglio, Jr., Table of n, a(n) for n = 0..100
David Consiglio, Jr., Python program

Position of first appearance of n in A001222 - A136565.
For factors instead of exponents we have A280286 (sorted A381075), firsts of A280292.
For indices instead of exponents we have A380956 (sorted A380957), firsts of A380955.
A000040 lists the primes, differences A001223.
A005361 gives product of prime exponents.
A055396 gives least prime index, greatest A061395.
A056239 (reverse A296150) adds up prime indices, row sums of A112798.
A124010 lists prime exponents (signature); A001221, A051903, A051904.
Cf. A046660, A071625, A075254, A075255, A076694, A130091, A130092, A290106, A380986.

prisig[n_]:=If[n==1,{},Last/@FactorInteger[n]];
q=Table[Total[prisig[n]]-Total[Union[prisig[n]]],{n,10000}];
mnrm[s_]:=If[Min@@s==1,mnrm[DeleteCases[s-1,0]]+1,0];
Table[Position[q,k][[1,1]],{k,0,mnrm[q+1]-1}]

a(10)-a(11) from Michel Marcus, Feb 20 2025

a(12) and beyond from David Consiglio, Jr., Feb 20 2025

cp's OEIS Frontend

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

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A383508 Number of integer partitions of n that are both Look-and-Say and section-sum partitions.

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A383510 Number of integer partitions of n that are neither Look-and-Say nor section-sum.

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A383511 Number of integer partitions of n that are Look-and-Say and section-sum but not Wilf.

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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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A380958 Number of prime factors of n (with multiplicity) minus sum of distinct prime exponents of n.

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A383518 Heinz numbers of integer partitions that are Look-and-Say and section-sum but not conjugate Wilf partitions.

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