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A357864
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Numbers whose prime indices have strictly decreasing run-sums. Heinz numbers of the partitions counted by A304430.
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9
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1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 24, 25, 27, 29, 31, 32, 37, 41, 43, 45, 47, 48, 49, 53, 59, 61, 64, 67, 71, 73, 79, 80, 81, 83, 89, 96, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 135, 137, 139, 149, 151, 157, 160, 163, 167, 169, 173
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OFFSET
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1,2
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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:
1: {}
2: {1}
3: {2}
4: {1,1}
5: {3}
7: {4}
8: {1,1,1}
9: {2,2}
11: {5}
13: {6}
16: {1,1,1,1}
17: {7}
19: {8}
23: {9}
24: {1,1,1,2}
25: {3,3}
27: {2,2,2}
29: {10}
For example, the prime indices of 24 are {1,1,1,2}, with run-sums (3,2), which are strictly decreasing, so 24 is in the sequence.
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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[300], Greater@@Total/@Split[primeMS[#]]&]
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CROSSREFS
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These partitions are counted by A304430.
These are the indices of rows in A354584 that are strictly decreasing.
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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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