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A322847
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Numbers whose prime indices have no equivalent primes.
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9
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1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 44, 45, 46, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 74, 75
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OFFSET
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1,2
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COMMENTS
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The complement is {13, 26, 29, 43, 47, 52, 58, 73, 79, 86, 94, ...}.
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.
In an integer partition, two primes are equivalent if each part has in its prime factorization the same multiplicity of both primes. For example, in (6,5) the primes {2,3} are equivalent while {2,5} and {3,5} are not. In (30,6) also, the primes {2,3} are equivalent, while {2,5} and {3,5} are not.
Also MM-numbers of T_0 multiset multisystems. A multiset multisystem is a finite multiset of finite multisets. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}. The dual of a multiset multisystem has, for each vertex, one block consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict (no repeated parts).
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LINKS
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EXAMPLE
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The prime indices of 339 are {2, 30}, in which the primes {3,5} are equivalent, so 339 is not 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}]]]];
dual[eds_]:=Table[First/@Position[eds, x], {x, Union@@eds}];
Select[Range[100], UnsameQ@@dual[primeMS/@primeMS[#]]&]
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CROSSREFS
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Cf. A056239, A059201, A112798, A302242, A316978, A316979, A316983, A319558, A319564, A319728, A322846.
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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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