Problems & Puzzles: Puzzles

Puzzle 441. σ(n)+π(n)=k.n JM Bergot asked recently for integers n such that: σ(n)+π(n)=k.n. ... , k integer too  (I) σ(n) = sum of divisors of n π(n) = prime counting function of n On my request, Farideh Firoozbakht computed a few n values satisfying (I) 2,3,15,22,98,21170,3587402,25881424,25888784 & 33305870 (less that 5x10^7) Q1. Do you devise an interesting property for these numbers? Q2. Are there more odd n numbers like these in the list (3 & 15)

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Contributions came from J. K. Andersen and Farideh Firoozbakht.

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JKA wrote:

The given solutions by Farideh Firoozbakht all have k=2.
n=1 is a trivial odd solution for k=1. There are no other odd solutions below 5*10^11. n=617086665 is a near miss with sigma(n) + pi(n) = 2*n + 1.

There are two other even solutions below 5*10^11: n=3902496 for k=3, and n=135135182 for k=2. There are many near misses. The largest is n=57115502744 with sigma(n) + pi(n) = 2*n + 1.

For very large random n, I guess sigma(n) + pi(n) behaves sufficiently randomly to have estimated chance around 1/n of being divisible by n. The harmonic series diverges, so I guess there are infinitely many solutions for both odd n and even n.

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Farideh wrote:

Note that 1 is in the sequence and next term of the sequence is
135135182. The only odd terms of the sequence up to 1.385*10^9 are 1, 3 & 15.

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