

A319899


Numbers whose number of prime factors with multiplicity (A001222) is the number of distinct prime factors (A001221) in the product of the prime indices (A003963).


2



1, 3, 5, 7, 11, 15, 17, 19, 23, 26, 31, 33, 35, 39, 41, 51, 53, 55, 58, 59, 65, 67, 69, 74, 77, 83, 85, 86, 87, 91, 93, 94, 95, 97, 103, 109, 111, 119, 122, 123, 127, 129, 131, 142, 146, 155, 157, 158, 161, 165, 169, 177, 178, 179, 183, 185, 187, 191, 201, 202
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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. The multiset multisystem with MMnumber 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 MMnumber 78 is {{},{1},{1,2}}. This sequence lists all MMnumbers of square multiset multisystems, meaning the number of edges is equal to the number of distinct vertices.


LINKS



EXAMPLE

The sequence of multiset multisystems whose MMnumbers belong to the sequence begins:
1: {}
3: {{1}}
5: {{2}}
7: {{1,1}}
11: {{3}}
15: {{1},{2}}
17: {{4}}
19: {{1,1,1}}
23: {{2,2}}
26: {{},{1,2}}
31: {{5}}
33: {{1},{3}}
35: {{2},{1,1}}
39: {{1},{1,2}}
41: {{6}}
51: {{1},{4}}
53: {{1,1,1,1}}
55: {{2},{3}}
58: {{},{1,3}}
59: {{7}}
65: {{2},{1,2}}
67: {{8}}
69: {{1},{2,2}}
74: {{},{1,1,2}}


MATHEMATICA

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
Select[Range[100], PrimeOmega[#]==PrimeNu[Times@@primeMS[#]]&]


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



