A316529 - OEIS

A316529

Heinz numbers of totally 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

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OFFSET

1,2

COMMENTS

First differs from A304678 at a(115) = 151, A304678(115) = 150.

The alternating version first differs from this sequence in having 150 and lacking 450.

An integer partition is totally strong if either it is empty, equal to (1), or its run-lengths are weakly decreasing (strong) and are themselves a totally strong partition.

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

Starting with (3,3,2,1), which has Heinz number 150, and repeatedly taking run-lengths gives (3,3,2,1) -> (2,1,1) -> (1,2), so 150 is not in the sequence.

Starting with (3,3,2,2,1), which has Heinz number 450, and repeatedly taking run-lengths gives (3,3,2,2,1) -> (2,2,1) -> (2,1) -> (1,1) -> (2) -> (1), so 450 is in the sequence.

MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

totstrQ[q_]:=Or[q=={}, q=={1}, And[GreaterEqual@@Length/@Split[q], totstrQ[Length/@Split[q]]]];

Select[Range[100], totstrQ[Reverse[primeMS[#]]]&]

CROSSREFS

Cf. A056239, A181819, A182850, A242031, A296150, A305732, A317246.

The enumeration of these partitions by sum is A316496.

The complement is A316597.

The widely normal version is A332291.

The dual version is A335376.

Partitions with weakly decreasing run-lengths are A100882.

Cf. A025487, A100883, A133808, A317257, A332275, A332293, A332297.