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A367683
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Numbers whose sorted prime signature is the same as the multiset multiplicity kernel of their prime indices.
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5
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1, 2, 6, 9, 10, 12, 14, 18, 22, 26, 30, 34, 38, 40, 42, 46, 58, 62, 66, 70, 74, 78, 82, 86, 90, 94, 102, 106, 110, 112, 114, 118, 122, 125, 126, 130, 134, 138, 142, 146, 154, 158, 166, 170, 174, 178, 182, 186, 190, 194, 198, 202, 206, 210, 214, 218, 222, 225
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OFFSET
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1,2
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COMMENTS
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We define the multiset multiplicity kernel (MMK) of a positive integer n to be the product of (least prime factor with exponent k)^(number of prime factors with exponent k) over all distinct exponents k appearing in the prime factorization of n. For example, 90 has prime factorization 2^1 * 3^2 * 5^1, so for k = 1 we have 2^2, and for k = 2 we have 3^1, so MMK(90) = 12. 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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Michael De Vlieger, 20 X 20 color coded list of terms, color function: gray = 1, red = prime, gold = composite prime power, green = squarefree composite, blue-purple = numbers neither squarefree nor prime powers. Bright green = primorial, light green = even squarefree semiprime, light blue = highly composite, middle blue = in A055932, purple = squareful but not a prime power.
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EXAMPLE
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The terms together with their prime indices begin:
1: {}
2: {1}
6: {1,2}
9: {2,2}
10: {1,3}
12: {1,1,2}
14: {1,4}
18: {1,2,2}
22: {1,5}
26: {1,6}
30: {1,2,3}
34: {1,7}
38: {1,8}
40: {1,1,1,3}
42: {1,2,4}
46: {1,9}
58: {1,10}
62: {1,11}
66: {1,2,5}
70: {1,3,4}
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MATHEMATICA
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mmk[q_]:=With[{mts=Length/@Split[q]}, Sort[Table[Min@@Select[q, Count[q, #]==i&], {i, mts}]]];
Select[Range[100], #==1||Sort[Last/@FactorInteger[#]] == mmk[PrimePi/@Join@@ConstantArray@@@FactorInteger[#]]&]
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CROSSREFS
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Partitions of this type are counted by A367682.
A367582 counts partitions by sum of multiset multiplicity kernel.
Cf. A000720, A000961, A005117, A051904, A052409, A071625, A072774, A130091, A367584, A367586, A367587.
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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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