A332290 Heinz numbers of widely alternately co-strongly normal integer partitions.

1, 2, 4, 6, 8, 12, 16, 30, 32, 60, 64, 128, 210, 256, 360, 512, 1024, 2048, 2310, 2520, 4096, 8192, 16384, 30030, 32768, 65536, 75600, 131072, 262144, 510510, 524288

Offset

1,2

Comments

An integer partition is widely alternately co-strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly increasing run-lengths (co-strong) which, if reversed, are themselves a widely alternately co-strongly normal partition.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

This sequence is closed under A181821, so there are infinitely many terms that are not powers of 2 or primorial numbers.

Links

Table of n, a(n) for n=1..31.

Example

The sequence of all widely alternately co-strongly normal integer partitions together with their Heinz numbers begins:
      1: ()
      2: (1)
      4: (1,1)
      6: (2,1)
      8: (1,1,1)
     12: (2,1,1)
     16: (1,1,1,1)
     30: (3,2,1)
     32: (1,1,1,1,1)
     60: (3,2,1,1)
     64: (1,1,1,1,1,1)
    128: (1,1,1,1,1,1,1)
    210: (4,3,2,1)
    256: (1,1,1,1,1,1,1,1)
    360: (3,2,2,1,1,1)
    512: (1,1,1,1,1,1,1,1,1)
   1024: (1,1,1,1,1,1,1,1,1,1)
   2048: (1,1,1,1,1,1,1,1,1,1,1)
   2310: (5,4,3,2,1)
   2520: (4,3,2,2,1,1,1)
For example, starting with y = (4,3,2,2,1,1,1), which has Heinz number 2520, and repeatedly taking run-lengths and reversing gives (4,3,2,2,1,1,1) -> (3,2,1,1) -> (2,1,1) -> (2,1) -> (1,1). These are all normal with weakly increasing run-lengths and the last is all 1's, so 2520 belongs to the sequence.

Mathematica

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
totnQ[ptn_]:=Or[ptn=={}, Union[ptn]=={1}, And[Union[ptn]==Range[Max[ptn]], LessEqual@@Length/@Split[ptn], totnQ[Reverse[Length/@Split[ptn]]]]];
Select[Range[10000], totnQ[Reverse[primeMS[#]]]&]

Crossrefs

Closed under A181821.

The non-co-strong version is A332276.

The enumeration of these partitions by sum is A332289.

The total (rather than alternating) version is A332293.

Cf. A055932, A056239, A100883, A133808, A181819, A317089, A317090, A317246, A317257, A317492, A329747, A332292, A332340.

Sequence in context: A344181 A242101 A131117 * A332293 A181821 A241882

Adjacent sequences: A332287 A332288 A332289 * A332291 A332292 A332293

Keyword

nonn,more

Author

Gus Wiseman, Feb 14 2020

Status

approved