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A182854 Integers whose prime signature a) contains at least two distinct numbers, and b) contains no number that occurs less often than any other number.

%I A182854 #10 Feb 16 2025 08:33:13
%S A182854 12,18,20,24,28,40,44,45,48,50,52,54,56,63,68,72,75,76,80,88,92,96,98,
%T A182854 99,104,108,112,116,117,124,135,136,144,147,148,152,153,160,162,164,
%U A182854 171,172,175,176,184,188,189,192,200,207,208,212,224,232,236,242,244,245
%N A182854 Integers whose prime signature a) contains at least two distinct numbers, and b) contains no number that occurs less often than any other number.
%C A182854 Numbers that require exactly four iterations to reach a fixed point under the x -> A181819(x) map. In each case, 2 is the fixed point that is reached. (1 is the other fixed point of the x -> A181819(x) map.) Cf. A182850.
%C A182854 Not the same sequence as A177425, which is a proper subsequence. 1260 is the first nonmember of A177425 that belongs to this sequence; its prime signature is (2,2,1,1).
%H A182854 Amiram Eldar, <a href="/A182854/b182854.txt">Table of n, a(n) for n = 1..10000</a>
%H A182854 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FixedPoint.html">Fixed Point</a>
%H A182854 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Map.html">Map</a>
%e A182854 The prime signature of 12 (2^2*3^1) is (2,1). Since (2,1) contains at least two distinct numbers, and since each number that appears in (2,1) appears exactly as often as any other number that appears, 12 belongs to this sequence.
%e A182854 12 also requires exactly four iterations under the x -> A181819(x) map to reach a fixed point (namely, 2) .  A181819(12) = 6;  A181819(6) = 4; A181819(4) = 3;  A181819(3) = 2 (and A181819(2) = 2).
%t A182854 aQ[n_] := Length[v = Values @ Counts @ FactorInteger[n][[;;,2]]] > 1 && Length @ Union @ v == 1; Select[Range[250], aQ] (* _Amiram Eldar_, Aug 08 2019 *)
%Y A182854 Numbers n such that A182850(n) = 4. See also A182853, A182855.
%Y A182854 Subsequence of A059404 and A182852.
%K A182854 nonn
%O A182854 1,1
%A A182854 _Matthew Vandermast_, Jan 04 2011