A370406 Primitive terms of A370348.

4, 18, 27, 50, 125, 225, 242, 294, 441, 578, 686, 1029, 1089, 1331, 1922, 2401, 2601, 3025, 3362, 3675, 4913, 5070, 5290, 6962, 7225, 7605, 8575, 8649, 8978, 12675, 13182, 13225, 13778, 15129, 15162, 17787, 19773, 21970, 22743, 23762, 23805, 24025, 24334, 29791, 31329, 32258, 32955, 34969, 35378

Offset: 1

Author

Robert Israel, Feb 17 2024

Keywords

• nonn

Comments

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

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.

Examples

			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.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..800

Crossrefs

Cf. A368110, A370348.

Programs

Maple

filter:= proc(n) uses numtheory; local F,D,t;
  if ormap(t -> n mod t = 0, S) then return false fi;
F:= map(t -> [pi(t[1]), t[2]], ifactors(n)[2]);
D:= `union`(seq(divisors(t[1]), t = F);
nops(D) < add(t[2], t = F);
end proc:
R:= NULL: count:= 0: S:= {}:
for n from 1 while count < 100 do
 if filter(n) then
   R:= R, n; S:= S union {n}; count:= count+1;
 fi
od:
R;

Mathematica

filter[n_] := Module[{F, d},
   If[AnyTrue[S, Mod[n, #] == 0&], Return[False]];
   F = {PrimePi[#[[1]]], #[[2]]} & /@ FactorInteger[n];
   d = Union[Flatten[Divisors /@ F[[All, 1]]]];
   Length[d] < Total[F[[All, 2]]]];
R = {}; count = 0; S = {};
For[n = 1, count < 100, n++, If[filter[n], AppendTo[R, n]; S = Union[S, {n}]; count++]];
R (* Jean-François Alcover, Mar 08 2024, after Robert Israel *)