A positive integer n is called an abundant number if (n) > 2n. Determine all abundant numbers

less than 50.

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- Aug 17th 2012, 12:54 AMmusngiburgerdetermining abundant number?
A positive integer n is called an abundant number if (n) > 2n. Determine all abundant numbers

less than 50. - Aug 17th 2012, 03:02 AMDevenoRe: determining abundant number?
start by "crossing numbers off the list". for example:

σ(p) = p+1 < 2p (if p is an odd prime).

σ(pq) = pq + p + q + 1 < pq + p + p + 1 = pq + 2p + 1 ≤ pq + pq = 2pq (when p,q are distinct odd primes with p > q)

(to see the last inequality, note that if q is an odd prime: q - 2 ≥ 1, so p(q - 2) ≥ 1, so pq ≥ 2p + 1).

σ(p^{2}) = p^{2}+ p + 1 < p^{2}+ 2p < p^{2}+ p^{2}= 2p^{2}(for any odd prime p).

can you think of some more ways to eliminate numbers without actually computing σ(n) for every single number?