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A367586
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Numbers whose prime indices have a multiset multiplicity kernel (in which each prime exponent becomes the least prime factor with that exponent) that is all ones {1,1,...}. Positions of powers of 2 in A367580.
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9
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1, 2, 4, 6, 8, 10, 14, 16, 22, 26, 30, 32, 34, 36, 38, 42, 46, 58, 62, 64, 66, 70, 74, 78, 82, 86, 94, 100, 102, 106, 110, 114, 118, 122, 128, 130, 134, 138, 142, 146, 154, 158, 166, 170, 174, 178, 182, 186, 190, 194, 196, 202, 206, 210, 214, 216, 218, 222
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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.
We define the multiset multiplicity kernel MMK(m) of a multiset m by the following property, holding for all distinct multiplicities k >= 1. If S is the set of elements of multiplicity k in m, then min(S) has multiplicity |S| in MMK(m). For example, MMK({1,1,2,2,3,4,5}) = {1,1,3,3,3}, and MMK({1,2,3,4,5,5,5,5}) = {1,1,1,1,5}. As an operation on multisets MMK is represented by A367579, and as an operation on their ranks it is represented by A367580.
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LINKS
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FORMULA
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Consists of 1 and all even terms of A072774 (powers of squarefree numbers).
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EXAMPLE
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We have MMK({1,1,2,2}) = {1,1} so 36 is in the sequence.
The terms together with their prime indices begin:
1: {}
2: {1}
4: {1,1}
6: {1,2}
8: {1,1,1}
10: {1,3}
14: {1,4}
16: {1,1,1,1}
22: {1,5}
26: {1,6}
30: {1,2,3}
32: {1,1,1,1,1}
34: {1,7}
36: {1,1,2,2}
38: {1,8}
42: {1,2,4}
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MATHEMATICA
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Select[Range[100], #==1||EvenQ[#]&&SameQ@@Last/@FactorInteger[#]&]
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CROSSREFS
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Partitions of this type (uniform containing 1) are counted by A097986.
Positions of all one rows {1,1,...} in A367579.
Positions of powers of 2 in A367580.
A071625 counts distinct prime exponents.
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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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