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

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

1,2

Comments

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..45.

Formula

The complement is A381433 U A381435.

Example

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}

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], Count[Times@@Prime/@#&/@egs/@IntegerPartitions[Total[prix[#]]], #]==1&]

Crossrefs

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.

Sequence in context: A246293 A047228 A032968 * A079279 A364381 A246444

Adjacent sequences: A381431 A381432 A381433 * A381435 A381436 A381437

Keyword

nonn,changed

Author

Gus Wiseman, Feb 27 2025

Status

approved