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

Offset: 1

Author

M. F. Hasler, May 31 2010

Keywords

• nonn

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.)

Examples

			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 ...
		

Crossrefs

Cf. A178483

Programs

PARI

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