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Math Help - determining abundant number?

  1. #1
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    determining abundant number?

    A positive integer n is called an abundant number if (n) > 2n. Determine all abundant numbers
    less than 50.
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  2. #2
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    Re: 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(p2) = p2 + p + 1 < p2 + 2p < p2 + p2 = 2p2 (for any odd prime p).

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