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%I #6 May 04 2019 03:40:25
%S 1,1,2,3,6,1,2,4,3,6,12,9,18,36,5,10,20,15,30,60,45,90,180,25,50,100,
%T 75,150,300,225,450,900,1,2,4,8,3,6,12,24,9,18,36,72,27,54,108,216,5,
%U 10,20,40,15,30,60,120,45,90,180,360,135,270,540,1080,25,50,100,200,75,150
%N For n=1,2,... list all products of the first n primes raised to some nonnegative power less than n.
%C Alternate construction: For n=1,2,... write all strings of length n using the first n symbols of an alphabet (a; aa,ab,ba,bb; aaa,aab,aac, aba,...), then code / interpret them as "positional" notation of exponents (a=0, b=1, ...) of primes (last digit = least prime), e.g.: bac => [1,0,2] => 5^1 3^0 2^2.
%C Obviously every natural numbers appears infinitely often (even after any other natural number) in this sequence. Thus any sequence of positive terms is a subsequence of this one.
%C A178484 is a more condensed version of this sequence.
%H Ivan Neretin, <a href="/A178483/b178483.txt">Table of n, a(n) for n = 1..10000</a>
%e The sequence begins: a(1)=2^0; a(2)=2^0 3^0, a(3)=2^1 3^0, a(4)=2^0 3^1, a(5)=2^1 3^1;
%e a(6,...)=2^0 3^0 5^0, 2^1 3^0 5^0, 2^2 3^0 5^0,
%e ________ 2^0 3^1 5^0, 2^1 3^1 5^0, 2^2 3^1 5^0,
%e ________ 2^0 3^2 5^0, 2^1 3^2 5^0, 2^2 3^2 5^0,
%e ________ 2^0 3^0 5^1, 2^1 3^0 5^1, 2^2 3^0 5^1,
%e ________ 2^0 3^1 5^1, 2^1 3^1 5^1, 2^2 3^1 5^1,
%e ________ 2^0 3^2 5^1, 2^1 3^2 5^1, 2^2 3^2 5^1,
%e ________ 2^0 3^0 5^2, 2^1 3^0 5^2, 2^2 3^0 5^2,
%e ________ 2^0 3^1 5^2, 2^1 3^1 5^2, 2^2 3^1 5^2,
%e ________ 2^0 3^2 5^2, 2^1 3^2 5^2, 2^2 3^2 5^2,...
%t {1}~Join~Flatten@Table[Times @@ (Prime@Range@n^Reverse@PadLeft[ IntegerDigits[#, n], n]) & /@ (Range[n^n] - 1), {n, 2, 4}] (* _Ivan Neretin_, May 02 2019 *)
%o (PARI) for( L=1,4, forvec( v=vector(L,i,[0,L-1]), print1( prod( j=1,L,prime(j)^v[L-j+1] )",")))
%Y Cf. A178480, A178484.
%K nonn
%O 1,3
%A _M. F. Hasler_, May 31 2010
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