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A370406 Primitive terms of A370348.

%I A370406 #14 Mar 08 2024 08:36:14
%S A370406 4,18,27,50,125,225,242,294,441,578,686,1029,1089,1331,1922,2401,2601,
%T A370406 3025,3362,3675,4913,5070,5290,6962,7225,7605,8575,8649,8978,12675,
%U A370406 13182,13225,13778,15129,15162,17787,19773,21970,22743,23762,23805,24025,24334,29791,31329,32258,32955,34969,35378
%N A370406 Primitive terms of A370348.
%C A370406 Terms of A370348 that are not divisible by any other term of A370348.
%C A370406 Numbers k such that there are fewer divisors of prime indices of k than there are prime indices of k, and no proper divisor of k has this property.
%H A370406 Robert Israel, <a href="/A370406/b370406.txt">Table of n, a(n) for n = 1..800</a>
%e A370406 a(4) = 50 is a term because the prime indices of 50 = 2*5^2 are 1, 2, 2, and there are 3 of these but only 2 divisors of prime indices, namely 1 and 2, and 50 is not divisible by any of the previous terms 4, 18 and 27 of the sequence.
%p A370406 filter:= proc(n) uses numtheory; local F,D,t;
%p A370406   if ormap(t -> n mod t = 0, S) then return false fi;
%p A370406 F:= map(t -> [pi(t[1]), t[2]], ifactors(n)[2]);
%p A370406 D:= `union`(seq(divisors(t[1]), t = F);
%p A370406 nops(D) < add(t[2], t = F);
%p A370406 end proc:
%p A370406 R:= NULL: count:= 0: S:= {}:
%p A370406 for n from 1 while count < 100 do
%p A370406  if filter(n) then
%p A370406    R:= R, n; S:= S union {n}; count:= count+1;
%p A370406  fi
%p A370406 od:
%p A370406 R;
%t A370406 filter[n_] := Module[{F, d},
%t A370406    If[AnyTrue[S, Mod[n, #] == 0&], Return[False]];
%t A370406    F = {PrimePi[#[[1]]], #[[2]]} & /@ FactorInteger[n];
%t A370406    d = Union[Flatten[Divisors /@ F[[All, 1]]]];
%t A370406    Length[d] < Total[F[[All, 2]]]];
%t A370406 R = {}; count = 0; S = {};
%t A370406 For[n = 1, count < 100, n++, If[filter[n], AppendTo[R, n]; S = Union[S, {n}]; count++]];
%t A370406 R (* _Jean-François Alcover_, Mar 08 2024, after _Robert Israel_ *)
%Y A370406 Cf. A368110, A370348.
%K A370406 nonn
%O A370406 1,1
%A A370406 _Robert Israel_, Feb 17 2024