Problems & Puzzles: Puzzles

Puzzle 824. Insecure numbers Fred Schneider sent the following nice puzzle:   Let's define the "insecurity" of a number n as the minimum number of distinct prime factors required of some m such that sigma(m) =n.  (The pun is that the more factors needed to "build it up", the more insecure it is) 3, 36, 4144, 720896, 139853824 are the minimum insecure numbers for levels 1 to 5. Q. Can you extend this sequence?   Explanation: sigma(2) = 3 3 can be sigma for 2, a prime.   So, we say Insecure(3) = 1 (meaning a minimum of one factor) sigma(2 * 11) = 36:   36 can be a sigma for a semi-prime (2 * 11) but it cannot for a prime (36 = 35 + 1 = 5*7 + 1).   So, we say Insecure(36) = 2 (meaning a min of two prime factors) sigma(3 * 13 * 73) = 4144  Details omitted but it can be shown that 4144 can not be a sigma for a semi-prime or a prime.  But, it clearly can for a sphenic number (aka product of three factors) = 3 * 13 * 73.  So, we say Insecure(4144) = 3 (meaning a minimum of 3 factors) The next two minimum terms for insecurity of 4 and 5 are: sigma (3 * 31 * 43 * 127) = 720896.    sigma(3 * 31 * 43 * 127 * 193) = 139853824 Other points: For clarity, semi-primes, sphenic numbers, etc must be square-free. Further, any number s such that sigma(n) = s that is not square-free does not factor into computing insecure(s).

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