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A353844
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Starting with the multiset of prime indices of n, repeatedly take the multiset of run-sums until you reach a squarefree number. This number is prime (or 1) iff n belongs to the sequence.
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5
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1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 40, 41, 43, 47, 49, 53, 59, 61, 63, 64, 67, 71, 73, 79, 81, 83, 84, 89, 97, 101, 103, 107, 109, 112, 113, 121, 125, 127, 128, 131, 137, 139, 144, 149, 151, 157, 163, 167, 169, 173, 179
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OFFSET
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1,2
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COMMENTS
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The run-sums transformation is described by Kimberling at A237685 and A237750.
The runs of a sequence are its maximal consecutive constant subsequences. For example, the runs of {1,1,1,2,2,3,4} are {1,1,1}, {2,2}, {3}, {4}, with sums {3,3,4,4}.
Note that the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k), so this sequence lists Heinz numbers of partitions whose run-sum trajectory reaches an empty set or singleton.
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LINKS
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EXAMPLE
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The terms together with their prime indices begin:
1: {} 25: {3,3} 64: {1,1,1,1,1,1}
2: {1} 27: {2,2,2} 67: {19}
3: {2} 29: {10} 71: {20}
4: {1,1} 31: {11} 73: {21}
5: {3} 32: {1,1,1,1,1} 79: {22}
7: {4} 37: {12} 81: {2,2,2,2}
8: {1,1,1} 40: {1,1,1,3} 83: {23}
9: {2,2} 41: {13} 84: {1,1,2,4}
11: {5} 43: {14} 89: {24}
12: {1,1,2} 47: {15} 97: {25}
13: {6} 49: {4,4} 101: {26}
16: {1,1,1,1} 53: {16} 103: {27}
17: {7} 59: {17} 107: {28}
19: {8} 61: {18} 109: {29}
23: {9} 63: {2,2,4} 112: {1,1,1,1,4}
The trajectory 60 -> 45 -> 35 ends in a nonprime number 35, so 60 is not in the sequence.
The trajectory 84 -> 63 -> 49 -> 19 ends in a prime number 19, so 84 is in the sequence.
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MATHEMATICA
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ope[n_]:=Times@@Prime/@Cases[If[n==1, {}, FactorInteger[n]], {p_, k_}:>PrimePi[p]*k];
Select[Range[100], #==1||PrimeQ[NestWhile[ope, #, !SquareFreeQ[#]&]]&]
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CROSSREFS
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A304442 counts partitions with all equal run-sums.
A353851 counts compositions with all equal run-sums, ranked by A353848.
A353832 represents the operation of taking run-sums of a partition.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
Cf. A005811, A073093, A130091, A181819, A182857, A304660, A325239, A325277, A353839, A353862, A353867.
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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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