%I #8 Oct 22 2020 11:45:39

%S 1,2,3,4,5,6,7,8,9,10,11,12,14,16,17,18,19,20,21,22,23,24,25,27,28,31,

%T 32,34,36,38,40,41,42,44,46,48,49,50,53,54,56,57,59,62,63,64,67,68,72,

%U 76,80,81,82,83,84,88,92,96,97,98,100,103,106,108,109,112

%N Numbers whose prime indices are powers of a common prime number.

%C A prime index of n is a number m such that prime(m) divides n.

%H Robert Israel, <a href="/A302593/b302593.txt">Table of n, a(n) for n = 1..10000</a>

%e Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.

%e 01: {}

%e 02: {{}}

%e 03: {{1}}

%e 04: {{},{}}

%e 05: {{2}}

%e 06: {{},{1}}

%e 07: {{1,1}}

%e 08: {{},{},{}}

%e 09: {{1},{1}}

%e 10: {{},{2}}

%e 11: {{3}}

%e 12: {{},{},{1}}

%e 14: {{},{1,1}}

%e 16: {{},{},{},{}}

%e 17: {{4}}

%e 18: {{},{1},{1}}

%e 19: {{1,1,1}}

%e 20: {{},{},{2}}

%e 21: {{1},{1,1}}

%e 22: {{},{3}}

%e 23: {{2,2}}

%e 24: {{},{},{},{1}}

%e 25: {{2},{2}}

%e 27: {{1},{1},{1}}

%e 28: {{},{},{1,1}}

%e 31: {{5}}

%e 32: {{},{},{},{},{}}

%e 34: {{},{4}}

%e 36: {{},{},{1},{1}}

%e 38: {{},{1,1,1}}

%e 40: {{},{},{},{2}}

%p filter:= proc(n) local F,q;

%p uses numtheory;

%p F:= map(pi, factorset(n));

%p nops(`union`(op(map(factorset,F)))) <= 1

%p end proc:

%p select(filter, [$1..200]); # _Robert Israel_, Oct 22 2020

%t primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Select[Range[100],SameQ@@Join@@primeMS/@primeMS[#]&]

%Y Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A047968, A281113, A295924, A301760, A302242, A302243.

%K nonn

%O 1,2

%A _Gus Wiseman_, Apr 10 2018