A178483 For n=1,2,... list all products of the first n primes raised to some nonnegative power less than n.

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

Offset: 1

Author

M. F. Hasler, May 31 2010

Keywords

• nonn

Comments

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.

Examples

			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,...
		

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A178480, A178484.

Programs

Mathematica

{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 *)

PARI

for( L=1,4, forvec( v=vector(L,i,[0,L-1]), print1( prod( j=1,L,prime(j)^v[L-j+1] )",")))