A178479 For n=0,1,2,... list all numbers not occurring earlier which can be written as product of the first n primes raised to some nonnegative power not exceeding n.

1, 2, 3, 4, 6, 9, 12, 18, 36, 5, 8, 10, 15, 20, 24, 25, 27, 30, 40, 45, 50, 54, 60, 72, 75, 90, 100, 108, 120, 125, 135, 150, 180, 200, 216, 225, 250, 270, 300, 360, 375, 450, 500, 540, 600, 675, 750, 900, 1000, 1080, 1125, 1350, 1500, 1800, 2250, 2700, 3000, 3375

Offset

1,2

Comments

Every positive integer occurs exactly once in this sequence, but depending on its largest prime factor, it may appear quite late with respect to larger numbers. E.g. prime(4)=7=a(65) appears after a(4^3)=27000=(2*3*5)^3, prime(5)=11=a(626) appears after a(5^4)=(2*3*5*7)^4=1944810000.

First A000169(n) terms are the divisors of A181555(n), and a(A000169(n))=A181555(n). [From Matthew Vandermast, Oct 31 2010]

Links

Table of n, a(n) for n=1..58.

Example

n=0, n=1 and n=2 give a(1)=1 (empty product), a(2)=2=prime(1)^1,
and a(3..9) = 3, 4, 6, 9, 12, 18, 36: numbers 2^a 3^b with a,b <= 2.
n=3 gives a(10..64) = 5, 8, 10, 12, 15, 18...: numbers 2^a 3^b 5^c not occurring earlier, with a,b,c <= 3.

Prog

(PARI) { s=[]; for( L=0, 3, a=[]; forvec( v=vector(L, i, [0, L]), setsearch( s, t=prod( j=1, L, prime(j)^v[L-j+1] )) & next; s=setunion(s, Set(t)); a=concat(a, t)); apply(x->print1(x", "), vecsort(a))) }

Crossrefs

Cf. A178480, A178483, A178484, A111791.

Sequence in context: A126011 A018256 A111791 * A057285 A057287 A056751

Adjacent sequences: A178476 A178477 A178478 * A178480 A178481 A178482

Keyword

nonn

Author

M. F. Hasler, May 31 2010

Status

approved