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A178483
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For n=1,2,... list all products of the first n primes raised to some nonnegative power less than n.
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4
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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, 75, 150, 300, 225, 450, 900, 1, 2, 4, 8, 3, 6, 12, 24, 9, 18, 36, 72, 27, 54, 108, 216, 5, 10, 20, 40, 15, 30, 60, 120, 45, 90, 180, 360, 135, 270, 540, 1080, 25, 50, 100, 200, 75, 150
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OFFSET
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1,3
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COMMENTS
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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.
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.
A178484 is a more condensed version of this sequence.
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LINKS
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EXAMPLE
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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;
a(6,...)=2^0 3^0 5^0, 2^1 3^0 5^0, 2^2 3^0 5^0,
________ 2^0 3^1 5^0, 2^1 3^1 5^0, 2^2 3^1 5^0,
________ 2^0 3^2 5^0, 2^1 3^2 5^0, 2^2 3^2 5^0,
________ 2^0 3^0 5^1, 2^1 3^0 5^1, 2^2 3^0 5^1,
________ 2^0 3^1 5^1, 2^1 3^1 5^1, 2^2 3^1 5^1,
________ 2^0 3^2 5^1, 2^1 3^2 5^1, 2^2 3^2 5^1,
________ 2^0 3^0 5^2, 2^1 3^0 5^2, 2^2 3^0 5^2,
________ 2^0 3^1 5^2, 2^1 3^1 5^2, 2^2 3^1 5^2,
________ 2^0 3^2 5^2, 2^1 3^2 5^2, 2^2 3^2 5^2,...
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MATHEMATICA
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{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 *)
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PROG
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(PARI) for( L=1, 4, forvec( v=vector(L, i, [0, L-1]), print1( prod( j=1, L, prime(j)^v[L-j+1] )", ")))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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