A334969 Heinz numbers of alternately strong integer partitions.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 46, 47, 49, 50, 51, 53, 54, 55, 57, 58, 59, 61, 62, 64, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 81, 82, 83

Offset

1,2

Comments

First differs from A304678 in lacking 450.

First differs from A316529 (the totally strong version) in having 150.

A sequence is alternately strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and, when reversed, are themselves an alternately strong sequence.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

Links

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

Example

The sequence does not contain 450, the Heinz number of (3,3,2,2,1), because, while the multiplicities are weakly decreasing, their reverse (1,2,2) does not have weakly decreasing multiplicities.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
altstrQ[q_]:=Or[q=={}, q=={1}, And[GreaterEqual@@Length/@Split[q], altstrQ[Reverse[Length/@Split[q]]]]];
Select[Range[100], altstrQ[Reverse[primeMS[#]]]&]

Crossrefs

The co-strong version is A317257.

The case of reversed partitions is (also) A317257.

The total version is A316529.

These partitions are counted by A332339.

Totally co-strong partitions are counted by A332275.

Alternately co-strong compositions are counted by A332338.

Cf. A000041, A100883, A181819, A182850, A182857, A304660, A305563, A316496, A317256, A332292, A332340.

Sequence in context: A304678 A316529 A329138 * A362619 A065200 A342526

Adjacent sequences: A334966 A334967 A334968 * A334970 A334971 A334972

Keyword

nonn

Author

Gus Wiseman, Jun 09 2020

Status

approved