Sum of Digits Sequence

Let $a_0, a_1, \dots$ be an integer sequence defined by:

$a_0 = 1$;
for $n \ge 1$, $a_n$ is the sum of the digits of all preceding terms.

The sequence starts with $1, 1, 2, 4, 8, 16, 23, 28, 38, 49, \dots$
You are given $a_{10^6} = 31054319$.
Find $a_{10^{15}}$.

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:

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):
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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