The Floor’s Revenge

Define $f_k(n) = \sum_{i=0}^n f_k(\lfloor\frac i k \rfloor)$ where $f_k(0) = 1$ and $\lfloor x \rfloor$ denotes the floor function.
For example, $f_5(10) = 18$, $f_7(100) = 1003$, and $f_2(10^3) = 264830889564$.
Find $(\sum_{k=2}^{10} f_k(10^{14})) \bmod (10^9+7)$.

This is a complex math problem that falls into the category of number theory, specifically involving sequences, sums, and modular arithmetic. However, we’re given a special function f_k(n) to work with, as well as a few example values.

Given the constraints involving floor function, and the pattern emerging from the given examples, it seems probable that f_k(n) is based on a triangular number calculation. Remember that triangular numbers represent the number of dots that can form an equilateral triangle, and the n-th triangular number is the number of dots in the sequence 1+2+3+…+n. For example, the 5th triangular number is 1+2+3+4+5=15.

This is very similar to the definition $f_k(n) = \sum_{i=0}^n f_k(\lfloor\frac i k \rfloor)$, if you think of each f_k(floor(i/k)) as representing one “step” in the summation. In other words, it seems like f_k(n) is counting up every k-th triangular number and adding them all up to n — hence, the need to use floor(i/k) to ensure that we land exactly on each k-th number and don’t overshoot.

So if f_k(n) essentially counts every k-th triangular number, up to n, then we can simplify the calculation and directly sum up these special “k-th triangular numbers” using the formula for the sum of first n triangular numbers, which is n*(n+1)*(n+2)/6.

This will allow us to find f_k(n) for every k from 2 to 10, and for n=10^14 (which is within the constraints of the problem statement). We can then add up all these values as instructed, then take the result modulo 10^9 + 7 as required.

To proceed, we need to utilize the triangular number summations to evaluate f_k(n) for k=2 to 10 and n=10^14, then add these values up and take the modulo.

To implement this solution, you would calculate f_k(n) for every k from 2 to 10 and n being 10^14 (within the problem’s constraints). Add all these values and then take the result modulo (10^9 + 7).

However, keep in mind that because of the massive size of the numbers involved (10^14 as input as well as results potentially going into the billions), direct computation may not be efficient or even feasible, due to both time and space constraints. A mathematical software package that can handle large calculations or a programming language with support for big integers might be needed.

A note is that this whole explanation and approach is based on interpretation of the problem as well as the patterns observed in the given examples. There isn’t a grounded theory or deterministic algorithm provided, as the problem seems to be more of an exploration or puzzle than a standard algorithmic or mathematical question.

I hope that clarified some of the concepts and gave you a general direction for the problem. This problem requires comprehending functions, recursion, floor function, series summation, closed form expression, and modular arithmetic which are studied in number theory and discrete math. These kind of problems typically appear in competitive programming and mathematical olympiads. This is not a standard problem or topic covered in regular school curriculum. If you continue studying these concepts, they will become more familiar and you will develop deeper understanding which will help you solve these kind of problems on your own.

More Answers:
Prime-Sum Numbers
Chromatic Conundrum
Faulhaber’s Formulas

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