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

1,2

Comments

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.

Links

Table of n, a(n) for n=1..67.

Example

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}

Mathematica

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[#]]], #]&]

Crossrefs

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.

Sequence in context: A043091 A031943 A023800 * A320340 A350838 A364347

Adjacent sequences: A381429 A381430 A381431 * A381433 A381434 A381435

Keyword

nonn

Author

Gus Wiseman, Feb 27 2025

Status

approved