cp's OEIS Frontend

A302593 Numbers whose prime indices are powers of a common prime number.

%I A302593 #8 Oct 22 2020 11:45:39
%S A302593 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 A302593 32,34,36,38,40,41,42,44,46,48,49,50,53,54,56,57,59,62,63,64,67,68,72,
%U A302593 76,80,81,82,83,84,88,92,96,97,98,100,103,106,108,109,112
%N A302593 Numbers whose prime indices are powers of a common prime number.
%C A302593 A prime index of n is a number m such that prime(m) divides n.
%H A302593 Robert Israel, <a href="/A302593/b302593.txt">Table of n, a(n) for n = 1..10000</a>
%e A302593 Entry A302242 describes a correspondence between positive integers and multiset multisystems. In this case it gives the following sequence of set systems.
%e A302593 01: {}
%e A302593 02: {{}}
%e A302593 03: {{1}}
%e A302593 04: {{},{}}
%e A302593 05: {{2}}
%e A302593 06: {{},{1}}
%e A302593 07: {{1,1}}
%e A302593 08: {{},{},{}}
%e A302593 09: {{1},{1}}
%e A302593 10: {{},{2}}
%e A302593 11: {{3}}
%e A302593 12: {{},{},{1}}
%e A302593 14: {{},{1,1}}
%e A302593 16: {{},{},{},{}}
%e A302593 17: {{4}}
%e A302593 18: {{},{1},{1}}
%e A302593 19: {{1,1,1}}
%e A302593 20: {{},{},{2}}
%e A302593 21: {{1},{1,1}}
%e A302593 22: {{},{3}}
%e A302593 23: {{2,2}}
%e A302593 24: {{},{},{},{1}}
%e A302593 25: {{2},{2}}
%e A302593 27: {{1},{1},{1}}
%e A302593 28: {{},{},{1,1}}
%e A302593 31: {{5}}
%e A302593 32: {{},{},{},{},{}}
%e A302593 34: {{},{4}}
%e A302593 36: {{},{},{1},{1}}
%e A302593 38: {{},{1,1,1}}
%e A302593 40: {{},{},{},{2}}
%p A302593 filter:= proc(n) local F,q;
%p A302593   uses numtheory;
%p A302593   F:= map(pi, factorset(n));
%p A302593   nops(`union`(op(map(factorset,F)))) <= 1
%p A302593 end proc:
%p A302593 select(filter, [$1..200]); # _Robert Israel_, Oct 22 2020
%t A302593 primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A302593 Select[Range[100],SameQ@@Join@@primeMS/@primeMS[#]&]
%Y A302593 Cf. A000961, A001222, A003963, A005117, A007716, A056239, A275024, A047968, A281113, A295924, A301760, A302242, A302243.
%K A302593 nonn
%O A302593 1,2
%A A302593 _Gus Wiseman_, Apr 10 2018