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Автор rossatron, история, 9 лет назад, По-английски

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 if ek is divisible by pk, otherwise ck = ek - 1

Example:

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

144 = 2432, so f(144) = 2432 - 1 = 2431 = 48, as 4 is divisible by 2, exponent of 2 remains same.

Now let

Example: F(10) = 1 + 1 + 4 + 1 + 1 + 1 + 4 + 3 + 1 = 17

Now I want to evaluate F(N) for a fairly large value of N, say 1014. Can I do it without factorizing each number?

It is more difficult than the previous one.

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Автор rossatron, история, 9 лет назад, По-английски

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?

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