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

Offset: 1

Author

Gus Wiseman, Jul 29 2018

Keywords

• nonn

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.

Examples

			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.
		

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.

Programs

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[#]]]&]

Extensions

Updated with corrected terminology by Gus Wiseman, Mar 08 2020