Problems & Puzzles: Puzzles

Puzzle 1136 Divisors functions and Fermat primes Sebastián Martín Ruiz sent the following puzzle: Let p a prime number consider F(p)=sigma0(p-1) and G(p)=sigma1(p-1) where sigma0(p-1) is d(p-1) the number of divisors of p-1 sigma1(p-1) is the sum of divisors of p-1 Prove: Q1) F(p) is prime if and only if G(p) is prime These are the cases that I have obtained       p         F(p)     G(p) q1=3            2         3 q2=5            3         7 q3=17          5         31 q4=65537    17       131071        ?         65537 ?     ? If we call the sequence of primes qn for which F(qn) and G(qn) are both prime, Q2. prove that we have F(qn)=q(n-1) Q3) The primes qn obtained are Fermat primes except 257. Why doesn't this appear? Q4) If Q2 is true, this suggests that there is a Fermat prime for which p-1 has 65537 divisors. Find it. qn is OEIS A249759 F is OEIS A004249 G is OEIS A249761

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During the week 24-30 June 2023, contributions came from Alejandro Casini, Emmanuel Vantieghem.

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Alejandro wrote:

Here’s a first analysis of the puzzle

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Emmanuel wrote:

Here is what I found about Puzzle 1136.

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