login
Primitive terms of A370348.
1

%I #14 Mar 08 2024 08:36:14

%S 4,18,27,50,125,225,242,294,441,578,686,1029,1089,1331,1922,2401,2601,

%T 3025,3362,3675,4913,5070,5290,6962,7225,7605,8575,8649,8978,12675,

%U 13182,13225,13778,15129,15162,17787,19773,21970,22743,23762,23805,24025,24334,29791,31329,32258,32955,34969,35378

%N Primitive terms of A370348.

%C Terms of A370348 that are not divisible by any other term of A370348.

%C 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 Robert Israel, <a href="/A370406/b370406.txt">Table of n, a(n) for n = 1..800</a>

%e 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 filter:= proc(n) uses numtheory; local F,D,t;

%p if ormap(t -> n mod t = 0, S) then return false fi;

%p F:= map(t -> [pi(t[1]), t[2]], ifactors(n)[2]);

%p D:= `union`(seq(divisors(t[1]), t = F);

%p nops(D) < add(t[2], t = F);

%p end proc:

%p R:= NULL: count:= 0: S:= {}:

%p for n from 1 while count < 100 do

%p if filter(n) then

%p R:= R, n; S:= S union {n}; count:= count+1;

%p fi

%p od:

%p R;

%t filter[n_] := Module[{F, d},

%t If[AnyTrue[S, Mod[n, #] == 0&], Return[False]];

%t F = {PrimePi[#[[1]]], #[[2]]} & /@ FactorInteger[n];

%t d = Union[Flatten[Divisors /@ F[[All, 1]]]];

%t Length[d] < Total[F[[All, 2]]]];

%t R = {}; count = 0; S = {};

%t For[n = 1, count < 100, n++, If[filter[n], AppendTo[R, n]; S = Union[S, {n}]; count++]];

%t R (* _Jean-François Alcover_, Mar 08 2024, after _Robert Israel_ *)

%Y Cf. A368110, A370348.

%K nonn

%O 1,1

%A _Robert Israel_, Feb 17 2024