Problems & Puzzles: Puzzles

Puzzle 542.- Sigma (x)=q^n

Farideh Firoozbakht sent the following puzzle:

Sigma(3^4) = 11^2 Sigma(20^2) = 31^2. Q1. Can you find other numbers n such that sigma(n) = q^k where q is an odd prime and k is greater than 1 ? Q2. Can you find another prime p such that sigma(p^m) = q^n where m & n are integers greater than 1 and q is prime ? _______________ Note that for Mersenne primes p sigma(p) is of the form 2^k.

Sigma(3^4) = 11^2 Sigma(20^2) = 31^2.

Q1. Can you find other numbers n such that sigma(n) = q^k where q is an odd prime and k is greater than 1 ?

Q2. Can you find another prime p such that sigma(p^m) = q^n where m & n are integers greater than 1 and q is prime ?

_______________ Note that for Mersenne primes p sigma(p) is of the form 2^k.

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