A089016 Largest n-round number.

2, 30, 1260, 60060, 2042040, 446185740, 25878772920, 7420738134810, 304250263527210, 52331045326680120, 9223346738827371150, 1922760350154212639070, 469153525437627883933080

Offset: 0

Author

Paul Boddington, Nov 04 2003

Keywords

• easy
• nonn

Comments

A positive integer m is said to be n-round if it is divisible by all primes p satisfying p^(n+1) < m, or equivalently if all positive integers t < m satisfying GCD(t,m)=1 are divisible by at most n primes (counting multiplicities). Using the fact that p_(t+1)<2*p_t (p_t the (t)th prime) it is easy to prove that there are only finitely many n-round numbers for each n. 1-round numbers are usually called very round (A048597).

Examples

			a(4)=2042040 as follows. Certainly it is 4-round since it is <= 19^5 and divisible by all primes < 19. Also it is > 17^5, hence the largest 4-round number must be a multiple of 510510 = 2.3.5.7.11.13.17. But no 4-round number can be > 19^5 (since it is easy to prove that if p is a prime >= 19 and q is the next prime after p then 2.3.5....p > q^5 ). Thus 2042040, being the largest multiple of 510510 which is <= 19^5, must be the largest 4-round number.
		

Links

• T. D. Noe, Table of n, a(n) for n=0..100

Crossrefs

Cf. A048597, A122936 (2-round numbers), A122937 (3-round numbers).

Programs

Mathematica

Table[k=1; While[prod=Times@@Prime[Range[k]]; prodT. D. Noe, Sep 21 2006 *)

Extensions

More terms from T. D. Noe, Sep 21 2006