

A322554


Numbers whose product of prime indices is a power of a squarefree number (A072774).


6



1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 72, 73, 76, 79, 80
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OFFSET

1,2


COMMENTS

The complement is {35, 37, 39, 45, 61, 65, ...}.
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 regular multiset multisystems, where regularity means all vertexdegrees are equal.


LINKS



EXAMPLE

Most small numbers belong to this sequence. However, the sequence of multiset multisystems whose MMnumbers do not belong to this sequence begins:
35: {{2},{1,1}}
37: {{1,1,2}}
39: {{1},{1,2}}
45: {{1},{1},{2}}
61: {{1,2,2}}
65: {{2},{1,2}}
69: {{1},{2,2}}
70: {{},{2},{1,1}}
71: {{1,1,3}}
74: {{},{1,1,2}}
75: {{1},{2},{2}}
77: {{1,1},{3}}
78: {{},{1},{1,2}}
87: {{1},{1,3}}
89: {{1,1,1,2}}
90: {{},{1},{1},{2}}
91: {{1,1},{1,2}}
95: {{2},{1,1,1}}
99: {{1},{1},{3}}


MATHEMATICA

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


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



