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A334969 Heinz numbers of alternately strong integer partitions. 0
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 (list; graph; refs; listen; history; text; internal format)
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 * A065200 A342526 A325361

Adjacent sequences:  A334966 A334967 A334968 * A334970 A334971 A334972

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jun 09 2020

STATUS

approved

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Last modified September 17 11:16 EDT 2021. Contains 347478 sequences. (Running on oeis4.)