

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

Table of n, a(n) for n=1..67.


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

Cf. A003963, A056239, A062503, A072774, A112798, A302242, A302491, A320325, A320698, A322553.
Sequence in context: A273883 A132145 A272284 * A330232 A273882 A277112
Adjacent sequences: A322551 A322552 A322553 * A322555 A322556 A322557


KEYWORD

nonn


AUTHOR

Gus Wiseman, Dec 15 2018


STATUS

approved



