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%I
%S 2,30,1260,60060,2042040,446185740,25878772920,7420738134810,
%T 304250263527210,52331045326680120,9223346738827371150,
%U 1922760350154212639070,469153525437627883933080
%N Largest n-round number.
%C 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).
%H T. D. Noe, <a href="/A089016/b089016.txt">Table of n, a(n) for n=0..100</a>
%e 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.
%t Table[k=1; While[prod=Times@@Prime[Range[k]]; prod<Prime[k+1]^(n+1), k++ ]; prod=prod/Prime[k]; prod*Floor[Prime[k]^(n+1)/prod], {n,0,100}] (* _T. D. Noe_, Sep 21 2006 *)
%Y Cf. A048597, A122936 (2-round numbers), A122937 (3-round numbers).
%K easy,nonn
%O 0,1
%A _Paul Boddington_, Nov 04 2003
%E More terms from _T. D. Noe_, Sep 21 2006
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