A332291 Heinz numbers of widely totally strongly normal integer partitions.

1, 2, 4, 6, 8, 16, 18, 30, 32, 64, 128, 210, 256, 450, 512, 1024, 2048, 2250, 2310, 4096, 8192, 16384, 30030, 32768, 65536, 131072, 262144, 510510, 524288

Offset

1,2

Comments

An integer partition is widely totally strongly normal if either it is constant 1's (wide) or it covers an initial interval of positive integers (normal) and has weakly decreasing run-lengths (strong) which are themselves a widely totally 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 A304660, so there are infinitely many terms that are not powers of 2 or primorial numbers.

Links

Table of n, a(n) for n=1..29.

Example

The sequence of all widely totally strongly normal integer partitions together with their Heinz numbers begins:
      1: ()
      2: (1)
      4: (1,1)
      6: (2,1)
      8: (1,1,1)
     16: (1,1,1,1)
     18: (2,2,1)
     30: (3,2,1)
     32: (1,1,1,1,1)
     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)
    450: (3,3,2,2,1)
    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)
   2250: (3,3,3,2,2,1)
   2310: (5,4,3,2,1)
   4096: (1,1,1,1,1,1,1,1,1,1,1,1)

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]], GreaterEqual@@Length/@Split[ptn], totnQ[Length/@Split[ptn]]]];
Select[Range[10000], totnQ[Reverse[primeMS[#]]]&]

Crossrefs

Closed under A304660.

The non-strong version is A332276.

The co-strong version is A332293.

The case of reversed partitions is (also) A332293.

Heinz numbers of normal partitions with decreasing run-lengths are A025487.

Cf. A055932, A056239, A181819, A242031, A317089, A317246, A317257, A317492, A329747, A332277, A332278, A332290, A332292, A332297, A332337.

Sequence in context: A058602 A133808 A160560 * A093109 A070034 A329798

Adjacent sequences: A332288 A332289 A332290 * A332292 A332293 A332294

Keyword

nonn,more

Author

Gus Wiseman, Feb 14 2020

Status

approved