Repeated Permutation

We define a permutation as an operation that rearranges the order of the elements $\{1, 2, 3, …, n\}$.
There are $n!$ such permutations, one of which leaves the elements in their initial order.
For $n = 3$ we have $3! = 6$ permutations:

$P_1 =$ keep the initial order
$P_2 =$ exchange the 1st and 2nd elements
$P_3 =$ exchange the 1st and 3rd elements
$P_4 =$ exchange the 2nd and 3rd elements
$P_5 =$ rotate the elements to the right
$P_6 =$ rotate the elements to the left

If we select one of these permutations, and we re-apply the same permutation repeatedly, we eventually restore the initial order.For a permutation $P_i$, let $f(P_i)$ be the number of steps required to restore the initial order by applying the permutation $P_i$ repeatedly.For $n = 3$, we obtain:

$f(P_1) = 1$ : $(1,2,3) \to (1,2,3)$
$f(P_2) = 2$ : $(1,2,3) \to (2,1,3) \to (1,2,3)$
$f(P_3) = 2$ : $(1,2,3) \to (3,2,1) \to (1,2,3)$
$f(P_4) = 2$ : $(1,2,3) \to (1,3,2) \to (1,2,3)$
$f(P_5) = 3$ : $(1,2,3) \to (3,1,2) \to (2,3,1) \to (1,2,3)$
$f(P_6) = 3$ : $(1,2,3) \to (2,3,1) \to (3,1,2) \to (1,2,3)$

Let $g(n)$ be the average value of $f^2(P_i)$ over all permutations $P_i$ of length $n$.$g(3) = (1^2 + 2^2 + 2^2 + 2^2 + 3^2 + 3^2)/3! = 31/6 \approx 5.166666667\mathrm e0$$g(5) = 2081/120 \approx 1.734166667\mathrm e1$$g(20) = 12422728886023769167301/2432902008176640000 \approx 5.106136147\mathrm e3$

Find $g(350)$ and write the answer in scientific notation rounded to $10$ significant digits, using a lowercase e to separate mantissa and exponent, as in the examples above.

To find the value of $a_{10^{15}}$, we can use dynamic programming to calculate the sequence iteratively. We can start by initializing the sequence with the first few terms and then iteratively calculate each term until we reach the desired term.

Here’s the Python code to solve this problem:

“`python
def sum_of_digits(n):
# Function to calculate the sum of digits of a number
sum = 0
while n > 0:
sum += n % 10
n //= 10
return sum

def find_an(an, target):
# Function to find the value of a_n
for i in range(target):
an.append(sum_of_digits(sum(an)))
return an

target = 10**15 # The target term we want to find
an = [1, 1] # Initialize the sequence with the first two terms

# Find the value of a_n iteratively until we reach the target term
if target < len(an): result = an[target] else: an = find_an(an, target) result = an[-1] print("The value of a_{0} is: {1}".format(target, result)) ``` Running this code will give you the value of $a_{10^{15}}$. However, be aware that finding $a_{10^{15}}$ might take a long time depending on your computer's processing power and memory.

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