A381437 -id:A381437 - cp's OEIS frontend

A381432 Heinz numbers of section-sum partitions. Union of A381431.

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 20, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 56, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A320340, A364347, A350838 in containing 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    5: {3}
    7: {4}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   11: {5}
   13: {6}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   17: {7}
   19: {8}
   20: {1,1,3}
   22: {1,5}
   23: {9}
   25: {3,3}
   26: {1,6}
   27: {2,2,2}

Partitions of this type are counted by A239455, complement A351293.
The conjugate is A351294, union of A048767 (parts A381440, fixed A048768, A217605).
Union of A381431 (parts A381436).
The complement is A381433, conjugate A351295.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A047966, A051903, A066328, A116861, A130091, A212166, A238745, A239964, A317081, A381437.

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

A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

Original entry on oeis.org

6, 12, 18, 21, 24, 30, 36, 42, 48, 54, 60, 63, 66, 70, 72, 78, 84, 90, 96, 102, 105, 108, 110, 114, 120, 126, 132, 138, 140, 144, 147, 150, 154, 156, 162, 165, 168, 174, 180, 186, 189, 192, 198, 204, 210, 216, 220, 222, 228, 231, 234, 238, 240, 246, 252, 258

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

First differs from A364348, A364537, A350845 in not containing 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			The terms together with their prime indices begin:
    6: {1,2}
   12: {1,1,2}
   18: {1,2,2}
   21: {2,4}
   24: {1,1,1,2}
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   48: {1,1,1,1,2}
   54: {1,2,2,2}
   60: {1,1,2,3}
   63: {2,2,4}
   66: {1,2,5}
   70: {1,3,4}
   72: {1,1,1,2,2}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
   96: {1,1,1,1,1,2}
  102: {1,2,7}
  105: {2,3,4}
  108: {1,1,2,2,2}

Partitions of this type are counted by A351293, complement A239455.
The conjugate is A351295, union of A048767 (parts A381440, fixed A048768, A217605).
The complement is A381432, union of A381431 (conjugate A351294, parts A381436).
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A051903, A066328, A116861, A130091, A181819, A238745, A239964, A317081, A381437.

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

A381431 Heinz number of the section-sum partition of the prime indices of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 5, 7, 8, 9, 7, 11, 10, 13, 11, 11, 16, 17, 15, 19, 14, 13, 13, 23, 20, 25, 17, 27, 22, 29, 13, 31, 32, 17, 19, 17, 25, 37, 23, 19, 28, 41, 17, 43, 26, 33, 29, 47, 40, 49, 35, 23, 34, 53, 45, 19, 44, 29, 31, 59, 26, 61, 37, 39, 64, 23, 19, 67, 38

Offset: 1

Gus Wiseman, Feb 26 2025

nonn

The image first differs from A320340, A364347, A350838 in containing a(150) = 65.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The section-sum partition (A381436) of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

			Prime indices of 180 are (3,2,2,1,1), with section-sum partition (6,3), so a(180) = 65.
The terms together with their prime indices begin:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
   7: {4}
  11: {5}
  10: {1,3}
  13: {6}
  11: {5}
  11: {5}
  16: {1,1,1,1}

The conjugate is A048767, union A351294, complement A351295, fix A048768 (count A217605).
Taking length instead of sum in the definition gives A238745, conjugate A181819.
Partitions of this type are counted by A239455, complement A351293.
The union is A381432, complement A381433.
Values appearing only once are A381434, more than once A381435.
These are the Heinz numbers of rows of A381436, conjugate A381440.
Greatest prime index of each term is A381437, counted by A381438.
A000040 lists the primes, differences A001223.
A003963 gives product of prime indices.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A005117, A047966, A051903, A066328, A091602, A116861, A130091, A212166, A380955.

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

A122111(a(n)) = A048767(n).

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 28 2025

Row-lengths are A051903.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The section-sum partition of a multiset or partition y is defined as follows: (1) determine and remember the sum of all distinct parts, (2) remove one instance of each distinct part, (3) repeat until no parts are left. The remembered values comprise the section-sum partition. For example, starting with (3,2,2,1,1) we get (6,3).
Equivalently, the k-th part of the section-sum partition is the sum of all (distinct) parts that appear at least k times. Compare to the definition of the conjugate of a partition, where we count parts >= k.
The conjugate of a section-sum partition is a Look-and-Say partition; see A048767, union A351294, count A239455.

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

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

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

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 28 2025

Row lengths are A066328.
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).
The conjugate of a Look-and-Say partition is a section-sum partition; see A381431, union A381432, count A239455.

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

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

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

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

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

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

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

In A381431:
- fixed points are A000961, A000005
- conjugate is A048767, fixed points A048768, A217605
- all numbers present are A381432, conjugate A351294
- numbers missing are A381433, conjugate A351295
- numbers appearing only once are A381434, conjugate A381540
- numbers appearing more than once are A381435 (this), conjugate A381541
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions, complement A351293.
A381436 lists section-sum partition of prime indices, conjugate A381440.
Set multipartitions: A050320, A089259, A116540, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A051903, A066328, A116861, A130091, A212166, A239964, A317081, A381437.

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

The complement is A381434 U A381433.

A381434 Numbers appearing only once in A381431 (section-sum partition of prime indices).

Original entry on oeis.org

1, 2, 3, 4, 8, 9, 10, 14, 15, 16, 20, 22, 27, 28, 32, 33, 35, 40, 44, 45, 50, 55, 56, 64, 75, 77, 80, 81, 88, 98, 99, 100, 112, 128, 130, 135, 160, 170, 175, 176, 182, 190, 195, 196, 200

Offset: 1

Gus Wiseman, Feb 27 2025

nonn

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

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    4: {1,1}
    8: {1,1,1}
    9: {2,2}
   10: {1,3}
   14: {1,4}
   15: {2,3}
   16: {1,1,1,1}
   20: {1,1,3}
   22: {1,5}
   27: {2,2,2}
   28: {1,1,4}
   32: {1,1,1,1,1}

In A381431:
- fixed points are A000961, A000005
- conjugate is A048767, fixed points A048768, A217605
- all numbers present are A381432, conjugate A351294
- numbers missing are A381433, conjugate A351295
- numbers appearing only once are A381434 (this), conjugate A381540
- numbers appearing more than once are A381435, conjugate A381541
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions, complement A351293.
A381436 lists section-sum partition of prime indices, conjugate A381440.
Set multipartitions: A050320, A089259, A116540, A296119, A318360, A318361.
Partition ideals: A300383, A317141, A381078, A381441, A381452, A381454.
Cf. A000720, A003557, A051903, A066328, A116861, A130091, A212166, A239964, A317081, A381437.

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

The complement is A381433 U A381435.

A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 2, 1, 0, 2, 3, 1, 0, 0, 3, 4, 1, 2, 0, 0, 4, 7, 2, 1, 0, 0, 0, 5, 9, 4, 1, 2, 0, 0, 0, 6, 13, 4, 4, 1, 0, 0, 0, 0, 8, 18, 6, 3, 2, 3, 0, 0, 0, 0, 10, 26, 9, 5, 2, 2, 0, 0, 0, 0, 0, 12, 32, 12, 8, 4, 2, 4, 0, 0, 0, 0, 0, 15

Offset: 1

Gus Wiseman, Mar 01 2025

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

			Triangle begins:
   1
   1  1
   1  0  2
   2  1  0  2
   3  1  0  0  3
   4  1  2  0  0  4
   7  2  1  0  0  0  5
   9  4  1  2  0  0  0  6
  13  4  4  1  0  0  0  0  8
  18  6  3  2  3  0  0  0  0 10
  26  9  5  2  2  0  0  0  0  0 12
  32 12  8  4  2  4  0  0  0  0  0 15
  47 16 11  4  3  2  0  0  0  0  0  0 18
  60 23 12  8  3  2  5  0  0  0  0  0  0 22
  79 27 20  7  9  4  3  0  0  0  0  0  0  0 27
 Row n = 9 counts the following partitions:
  (711)        (522)    (333)     (441)  .  .  .  .  (9)
  (6111)       (4221)   (3321)                       (81)
  (5211)       (3222)   (32211)                      (72)
  (51111)      (22221)  (222111)                     (63)
  (4311)                                             (621)
  (42111)                                            (54)
  (411111)                                           (531)
  (33111)                                            (432)
  (321111)
  (3111111)
  (2211111)
  (21111111)
  (111111111)

Last column (k=n) is A000009.
Row sums are A000041.
Row sums without the last column (k=n) are A047967.
For first instead of last part we have A116861, rank A066328.
First column (k=1) is A241131 shifted right and starting with 1 instead of 0.
Using Heinz numbers, this statistic is given by A381437.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts section-sum partitions, complement A351293.
Set multipartitions: A050320, A089259, A116540, A270995, A296119, A318360, A318361.
Section-sum partition: A381431, A381432, A381433, A381434, A381435, A381436.
Look-and-Say partition: A048767, A351294, A351295, A381440.
Cf. A047966, A051903, A051904, A091602, A181819, A212166.

egs[y_]:=If[y=={},{},Table[Total[Select[Union[y],Count[y,#]>=i&]],{i,Max@@Length/@Split[y]}]];
Table[Length[Select[IntegerPartitions[n],k==Last[egs[#]]&]],{n,15},{k,n}]

A381439 Numbers whose exponent of 2 in their canonical prime factorization is not larger than all the other exponents.

Original entry on oeis.org

3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87, 89

Offset: 1

Gus Wiseman, Mar 02 2025

nonn

First differs from A335740 in lacking 72, which has prime indices {1,1,1,2,2} and section-sum partition (3,3,1).
Also numbers whose section-sum partition of prime indices (A381436) ends with a number > 1.
Includes all squarefree numbers (A005117) except 2.

			The terms together with their prime indices begin:
     3: {2}        25: {3,3}        45: {2,2,3}
     5: {3}        26: {1,6}        46: {1,9}
     6: {1,2}      27: {2,2,2}      47: {15}
     7: {4}        29: {10}         49: {4,4}
     9: {2,2}      30: {1,2,3}      50: {1,3,3}
    10: {1,3}      31: {11}         51: {2,7}
    11: {5}        33: {2,5}        53: {16}
    13: {6}        34: {1,7}        54: {1,2,2,2}
    14: {1,4}      35: {3,4}        55: {3,5}
    15: {2,3}      36: {1,1,2,2}    57: {2,8}
    17: {7}        37: {12}         58: {1,10}
    18: {1,2,2}    38: {1,8}        59: {17}
    19: {8}        39: {2,6}        61: {18}
    21: {2,4}      41: {13}         62: {1,11}
    22: {1,5}      42: {1,2,4}      63: {2,2,4}
    23: {9}        43: {14}         65: {3,6}

The LHS (exponent of 2) is A007814.
The complement is A360013 = 2*A360015 (if we prepend 1), counted by A241131 (shifted right and starting with 1 instead of 0).
The case of equality is A360014, inclusive A360015.
The RHS (greatest exponent of an odd prime factor) is A375669.
These are positions of terms > 1 in A381437.
Partitions of this type are counted by A381544.
A000040 lists the primes, differences A001223.
A051903 gives greatest prime exponent, least A051904.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
A122111 represents conjugation in terms of Heinz numbers.
A239455 counts Look-and-Say partitions, complement A351293.
A381436 gives section-sum partition of prime indices, Heinz number A381431.
A381438 counts partitions by last part part of section-sum partition.
Cf. A000720, A001221, A001222, A001694, A003557, A005117, A066328, A130091, A181819, A212166, A380955.

Select[Range[100],FactorInteger[2*#][[1,2]]-1<=Max@@Last/@Rest[FactorInteger[2*#]]&]

Positive integers n such that A007814(n) <= A375669(n).

A381544 Number of integer partitions of n not containing more ones than any other part.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 7, 8, 13, 17, 24, 30, 45, 54, 75, 97, 127, 160, 212, 263, 342, 427, 541, 672, 851, 1046, 1307, 1607, 1989, 2428, 2993, 3631, 4443, 5378, 6533, 7873, 9527, 11424, 13752, 16447, 19701, 23470, 28016, 33253, 39537, 46801, 55428, 65408, 77238

Offset: 0

Gus Wiseman, Mar 24 2025

nonn

			The a(2) = 1 through a(9) = 17 partitions:
  (2)  (3)   (4)   (5)    (6)     (7)     (8)      (9)
       (21)  (22)  (32)   (33)    (43)    (44)     (54)
             (31)  (41)   (42)    (52)    (53)     (63)
                   (221)  (51)    (61)    (62)     (72)
                          (222)   (322)   (71)     (81)
                          (321)   (331)   (332)    (333)
                          (2211)  (421)   (422)    (432)
                                  (2221)  (431)    (441)
                                          (521)    (522)
                                          (2222)   (531)
                                          (3221)   (621)
                                          (3311)   (3222)
                                          (22211)  (3321)
                                                   (4221)
                                                   (22221)
                                                   (32211)
                                                   (222111)

The complement is counted by A241131, ranks A360013 = 2*A360015 (if we prepend 1).
The Heinz numbers of these partitions are A381439.
The case of equality is A382303, ranks A360014.
A000041 counts integer partitions, strict A000009.
A008284 counts partitions by length, strict A008289.
A047993 counts partitions with max part = length, ranks A106529.
A091602 counts partitions by the greatest multiplicity, rank statistic A051903.
A116598 counts ones in partitions, rank statistic A007814.
A239964 counts partitions with max multiplicity = length, ranks A212166.
A240312 counts partitions with max part = max multiplicity, ranks A381542.
A382302 counts partitions with max = max multiplicity = distinct length, ranks A381543.
Cf. A047966, A091605, A116861, A232697, A237984, A362608, A363724, A381079, A381437, A381438.

Table[Length[Select[IntegerPartitions[n],Count[#,1]<=Max@@Length/@Split[DeleteCases[#,1]]&]],{n,0,30}]

cp's OEIS Frontend

A381432 Heinz numbers of section-sum partitions. Union of A381431.

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A381433 Heinz numbers of non section-sum partitions. Complement of A381431.

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A381431 Heinz number of the section-sum partition of the prime indices of n.

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

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

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

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A381434 Numbers appearing only once in A381431 (section-sum partition of prime indices).

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A381438 Triangle read by rows where T(n>0,k>0) is the number of integer partitions of n whose section-sum partition ends with k.

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A381439 Numbers whose exponent of 2 in their canonical prime factorization is not larger than all the other exponents.

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