

A357863


Numbers whose prime indices do not have strictly increasing runsums. Heinz numbers of the partitions not counted by A304428.


3



12, 24, 40, 45, 48, 60, 63, 80, 84, 90, 96, 112, 120, 126, 132, 135, 144, 156, 160, 168, 175, 180, 189, 192, 204, 224, 228, 240, 252, 264, 270, 275, 276, 280, 288, 297, 300, 312, 315, 320, 325, 336, 348, 350, 351, 352, 360, 372, 378, 384, 405, 408, 420, 440
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OFFSET

1,1


COMMENTS

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
The Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.
The sequence of runs of a sequence consists of its maximal consecutive constant subsequences when read lefttoright. For example, the runs of (2,2,1,1,1,3,2,2) are (2,2), (1,1,1), (3), (2,2), with sums (4,3,3,4).


LINKS



EXAMPLE

The terms together with their prime indices begin:
12: {1,1,2}
24: {1,1,1,2}
40: {1,1,1,3}
45: {2,2,3}
48: {1,1,1,1,2}
60: {1,1,2,3}
63: {2,2,4}
80: {1,1,1,1,3}
84: {1,1,2,4}
90: {1,2,2,3}
96: {1,1,1,1,1,2}
112: {1,1,1,1,4}
120: {1,1,1,2,3}
126: {1,2,2,4}
132: {1,1,2,5}
135: {2,2,2,3}
144: {1,1,1,1,2,2}
156: {1,1,2,6}


MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], !Less@@Total/@Split[primeMS[#]]&]


CROSSREFS

These are the indices of rows in A354584 that are not strictly increasing.
The weak (not weakly increasing) version is A357876, counted by A357878.


KEYWORD

nonn


AUTHOR



STATUS

approved



