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A370348 Numbers k such that there are fewer divisors of prime indices of k than there are prime indices of k.

%I A370348 #16 Mar 22 2024 16:28:12
%S A370348 4,8,12,16,18,20,24,27,32,36,40,44,48,50,54,56,60,64,68,72,80,81,84,
%T A370348 88,90,96,100,108,112,120,124,125,126,128,132,135,136,144,150,160,162,
%U A370348 164,168,176,180,184,189,192,196,198,200,204,208,216,220,224,225,236,240,242,243,248,250,252,256
%N A370348 Numbers k such that there are fewer divisors of prime indices of k than there are prime indices of k.
%C A370348 No multiple of a term is a term of A368110.
%H A370348 Robert Israel, <a href="/A370348/b370348.txt">Table of n, a(n) for n = 1..10000</a>
%e A370348 a(5) = 18 is a term because the prime indices of 18 = 2 * 3^2 are 1,2,2, and there are 3 of these but only 2 divisors of prime indices, namely 1 and 2.
%p A370348 filter:= proc(n) uses numtheory; local F,D,t;
%p A370348    F:= map(t -> [pi(t[1]),t[2]], ifactors(n)[2]);
%p A370348    D:= `union`(seq(divisors(t[1]), t = F));
%p A370348    nops(D) < add(t[2], t = F)
%p A370348 end proc:
%p A370348 select(filter, [$1..300]);
%t A370348 filter[n_] := Module[{F, d},
%t A370348     F = {PrimePi[#[[1]]], #[[2]]}& /@ FactorInteger[n];
%t A370348     d = Union[Flatten[Divisors /@ F[[All, 1]]]];
%t A370348     Length[d] < Total[F[[All, 2]]]];
%t A370348 Select[Range[300], filter] (* _Jean-François Alcover_, Mar 08 2024, after Maple code *)
%Y A370348 The LHS is A370820, firsts A371131.
%Y A370348 The version for equality is A370802, counted by A371130, strict A371128.
%Y A370348 For submultisets instead of parts on the RHS we get A371167.
%Y A370348 The opposite version is A371168, counted by A371173.
%Y A370348 The weak version is A371169.
%Y A370348 The complement is A371170.
%Y A370348 Partitions of this type are counted by A371171.
%Y A370348 A000005 counts divisors.
%Y A370348 A001221 counts distinct prime factors.
%Y A370348 A027746 lists prime factors, indices A112798, length A001222.
%Y A370348 A355731 counts choices of a divisor of each prime index, firsts A355732.
%Y A370348 Cf. A003963, A239312, A303975, A319899, A355529, A355739, A370808, A371127.
%K A370348 nonn
%O A370348 1,1
%A A370348 _Robert Israel_, Feb 15 2024