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A332293
Heinz numbers of widely totally co-strongly normal integer partitions.
7
1, 2, 4, 6, 8, 12, 16, 30, 32, 64, 128, 180, 210, 256, 360, 512, 1024, 2048, 2310, 4096, 8192, 16384, 30030, 32768, 65536, 75600, 131072, 262144, 510510, 524288
OFFSET
1,2
COMMENTS
An integer partition is widely totally 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 are themselves a widely totally co-strongly normal partition.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
12: {1,1,2}
16: {1,1,1,1}
30: {1,2,3}
32: {1,1,1,1,1}
64: {1,1,1,1,1,1}
128: {1,1,1,1,1,1,1}
180: {1,1,2,2,3}
210: {1,2,3,4}
256: {1,1,1,1,1,1,1,1}
360: {1,1,1,2,2,3}
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: {1,2,3,4,5}
4096: {1,1,1,1,1,1,1,1,1,1,1,1}
8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
For example, 180 is the Heinz number of (3,2,2,1,1), with run-lengths (3,2,2,1,1) -> (1,2,2) -> (1,2) -> (1,1). These are all normal with weakly increasing multiplicities and the last is all 1's, so 180 belongs to the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
gnaQ[y_]:=Or[y=={}, Union[y]=={1}, And[normQ[y], LessEqual@@Length/@Split[y], gnaQ[Length/@Split[y]]]];
Select[Range[1000], gnaQ[Reverse[primeMS[#]]]&]
CROSSREFS
A subset of A055932.
Closed under A181819.
The non-co-strong version is A332276.
The enumeration of these partitions by sum is A332278.
The alternating version is A332290.
The strong version is A332291.
The case of reversed partitions is (also) A332291.
Sequence in context: A242101 A131117 A332290 * A181821 A241882 A238626
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Feb 16 2020
STATUS
approved