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A343013 Lexicographically earliest strictly increasing sequence of numbers whose partial products have mutually distinct exponents in their prime factorization (A130091).

%I A343013 #10 Apr 03 2021 08:42:48
%S A343013 1,2,4,5,8,9,12,15,16,17,18,20,24,25,27,30,32,34,35,36,40,45,48,49,50,
%T A343013 52,54,56,60,63,64,68,70,72,75,78,79,80,81,84,85,90,91,96,98,100,102,
%U A343013 104,105,108,112,119,120,121,125,126,128,130,132,135,136,140,143
%N A343013 Lexicographically earliest strictly increasing sequence of numbers whose partial products have mutually distinct exponents in their prime factorization (A130091).
%C A343013 The numbers of terms not exceeding 10^k, for k=1,2,..., are 6, 46, 293, 1939, 13534, 97379, .... Apparently, this sequence has an asymptotic density 0.
%C A343013 Are there infinitely many terms of each prime signature? In particular, the prime terms seem to be sparse: 2, 5, 17, 79, 491, 2011, 8191 and no other below 10^6. Are there infinitely many prime terms in this sequence?
%H A343013 Amiram Eldar, <a href="/A343013/b343013.txt">Table of n, a(n) for n = 1..10000</a>
%e A343013 The first partial products are:
%e A343013 1
%e A343013 1 * 2 = 2 = 2^1
%e A343013 1 * 2 * 4 = 8 = 2^3
%e A343013 1 * 2 * 4 * 5 = 40 = 2^3 * 5^1
%e A343013 1 * 2 * 4 * 5 * 8 = 320 = 2^6 * 5^1
%t A343013 q[n_] := UnsameQ @@ FactorInteger[n][[;; , 2]]; seq = {1}; n = 1; prod = 1; Do[k = n + 1; While[!q[k*prod], k++]; AppendTo[seq, k]; prod *= k; n = k, {100}]; seq
%Y A343013 Cf. A130091, A343012.
%K A343013 nonn
%O A343013 1,2
%A A343013 _Amiram Eldar_, Apr 02 2021