Summation of a weird function.

Правка en3, от rossatron, 2015-06-26 17:36:04

I found this question at http://math.stackexchange.com/.

Let n is a positive integer.

n = p1e1p2e2... pkek is the complete prime factorization of n.

Let me define a function f(n)

f(n) = p1c1p2c2... pkck where ck = ek - 1

Example:

72 = 2332, so f(72) = 23 - 132 - 1 = 2231 = 12

144 = 2432, so f(144) = 24 - 132 - 1 = 2331 = 24

Now let

Example: F(10) = 1 + 1 + 2 + 1 + 1 + 1 + 4 + 3 + 1 = 15

Now I want to evaluate F(N) for a fairly large value of N, say 1012. Can I do it without factorizing each number?

It appears quite interesting. I think maybe it requires some kind of inclusion-exclusion, but so far was unable to implement. Any idea how to solve?

Теги number theory, summation, factorisation

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en3 Английский rossatron 2015-06-26 17:36:04 96
en2 Английский rossatron 2015-06-26 17:27:43 0 (published)
en1 Английский rossatron 2015-06-26 17:26:39 810 Initial revision (saved to drafts)