A178480 - OEIS

A178480

For n=0,1,2,... list all products of the first n primes raised to some positive power not exceeding n.

1, 2, 6, 12, 18, 36, 30, 60, 120, 90, 180, 360, 270, 540, 1080, 150, 300, 600, 450, 900, 1800, 1350, 2700, 5400, 750, 1500, 3000, 2250, 4500, 9000, 6750, 13500, 27000, 210, 420, 840, 1680, 630, 1260, 2520, 5040, 1890, 3780, 7560, 15120, 5670, 11340, 22680

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

Alternate construction: For n=0,1,2,... write all strings of length n using the first n symbols of the alphabet (""; a; aa,ab,ba,bb; aaa,aab,aac, aba,...), then code / interpret them as "positional" notation of exponents (a=1, b=2, ...) of primes (last digit = least prime), e.g.: acb => [1,3,2] => 5^1 3^3 2^2.

These numbers have the property that, if a prime p divides the number, then all primes less than p also divide it. (But not all such numbers are listed, neither are they listed in increasing order.)

LINKS

Table of n, a(n) for n=1..48.

EXAMPLE

The sequence begins: a(1)=1 (empty product); a(2)=2^1;

a(3,...,6)=2^1 3^1, 2^2 3^1, 2^1 3^2, 2^2 3^2;

a(7,...)=2^1 3^1 5^1, 2^2 3^1 5^1, 2^3 3^1 5^1,

________ 2^1 3^2 5^1, 2^2 3^2 5^1, 2^3 3^2 5^1,

________ 2^1 3^3 5^1, 2^2 3^3 5^1, 2^3 3^3 5^1,

________ 2^1 3^1 5^2, 2^2 3^1 5^2, 2^3 3^1 5^2, ...

They correspond to the strings (cf. comment) "" a aa ab ba bb aaa aab aac aba abb abc aca acb acc baa bab bac ...

PROG

(PARI) for( L=0, 3, forvec( v=vector(L, i, [1, L]), print1( prod( j=1, L, prime(j)^v[L-j+1] )", ")))