- #1

Guffel

- 28

- 0

Form the difference [tex]d_n[/tex] between the sum of the squares of the prime factors of n and n itself. For example, 44 = (2)(2)(11) => d44 = 2^2+2^2+11^2 - 44 = 85.

I've made calculations up to n=1,000,000.

Questions

1. I only found three values for n where [tex]d_n=0[/tex] (1, 16 and 27). Are these the only numbers that have this property? Proof?

2. The ratio between the number of values having the property that [tex]d_n<=0[/tex] and n seems to grow asymptotically to approximately 0.265. Any way to find this limit (or to prove that it is > 0)?

3. The number series formed by all n having the property that [tex]d_n<=0[/tex] starts with 1, 16, 24, 27, 32, 36, 40 and is similar to A166319 for all the values listed on oeis.org. The definition of the series is related, but not similar, to the one in this post. Is it obvious that they are similar?

Any input is welcome!