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A357863
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Numbers whose prime indices do not have strictly increasing run-sums. Heinz numbers of the partitions not counted by A304428.
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3
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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
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1,1
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COMMENTS
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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 left-to-right. 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).
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LINKS
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EXAMPLE
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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}
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MATHEMATICA
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primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], !Less@@Total/@Split[primeMS[#]]&]
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CROSSREFS
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These are the indices of rows in A354584 that are not strictly increasing.
The weak (not weakly increasing) version is A357876, counted by A357878.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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