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A322846 Squarefree numbers whose prime indices have no equivalent primes. 1
1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 46, 51, 53, 55, 57, 59, 61, 62, 65, 66, 67, 69, 70, 71, 74, 77, 78, 82, 83, 85, 87, 89, 91, 93, 95, 97, 102, 103, 105, 106, 107, 109, 110, 111, 114, 115, 118, 119 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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 strict 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).

LINKS

Table of n, a(n) for n=1..63.

EXAMPLE

The sequence of all strict T_0 multiset multisystems together with their MM-numbers begins:

   1: {}

   2: {{}}

   3: {{1}}

   5: {{2}}

   6: {{},{1}}

   7: {{1,1}}

  10: {{},{2}}

  11: {{3}}

  14: {{},{1,1}}

  15: {{1},{2}}

  17: {{4}}

  19: {{1,1,1}}

  21: {{1},{1,1}}

  22: {{},{3}}

  23: {{2,2}}

  30: {{},{1},{2}}

  31: {{5}}

  33: {{1},{3}}

  34: {{},{4}}

  35: {{2},{1,1}}

  37: {{1,1,2}}

  38: {{},{1,1,1}}

  39: {{1},{1,2}}

MATHEMATICA

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], And[SquareFreeQ[#], UnsameQ@@dual[primeMS/@primeMS[#]]]&]

CROSSREFS

Cf. A000009, A005117, A056239, A059201, A112798, A302242, A302505, A316978, A316979, A316983, A319558, A319564, A319728, A322847.

Sequence in context: A137313 A246867 A028805 * A302496 A117344 A117204

Adjacent sequences:  A322843 A322844 A322845 * A322847 A322848 A322849

KEYWORD

nonn

AUTHOR

Gus Wiseman, Dec 28 2018

STATUS

approved

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Last modified June 12 14:45 EDT 2021. Contains 344957 sequences. (Running on oeis4.)