#### Problem

The prime 41, can be written as the sum of six consecutive primes:

41 = 2 + 3 + 5 + 7 + 11 + 13

This is the longest sum of consecutive primes that adds to a prime below one-hundred.

The longest sum of consecutive primes below one-thousand that adds to a prime, contains 21 terms, and is equal to 953.

Which prime, below one-million, can be written as the sum of the most consecutive primes?

#### Solution

// generate all prime numbers under <= this max let max = 10000 // initialise the list with 2 which is the only even number in the sequence let mutable primeNumbers = [2] // only check the prime numbers which are <= the square root of the number n let hasDivisor n = primeNumbers |> Seq.takeWhile (fun n' -> n' <= int(sqrt(double(n)))) |> Seq.exists (fun n' -> n % n' = 0) // only check odd numbers <= max let potentialPrimes = Seq.unfold (fun n -> if n > max then None else Some(n, n+2)) 3 // populate the prime numbers list for n in potentialPrimes do if not(hasDivisor n) then primeNumbers Seq.filter (fun n' -> n' > n) |> Seq.scan (fun (sum, count) n' -> (sum+n', count+1)) (n, 1) |> Seq.takeWhile (fun (sum, count) -> sum < 1000000) |> Seq.filter (fun (sum, count) -> isPrime sum) |> Seq.maxBy (fun (sum, count) -> count) // for all numbers, find the longest sequence let answer = primeNumbers |> Seq.map getPrimeSequence |> Seq.maxBy (fun (sum, count) -> count)

Again, utilizing the *PGsimple 3* algorithm from here, this solution runs in just over a second as I’ve restricted it to check sequences starting with a prime < 10000 which perhaps unsurprisingly, has provided me with the answer.

The only tricky part in this problem is how to build up a list of sequences for each starting prime and find the longest sequence starting from that number which results in a prime < 1000000. Fortunately you can do this rather easily with Seq.scan in F# which works similarly to Seq.fold but gives you all the intermediate results as well as the final result. Used in conjunction with Seq.takeWhile it allows me to get all the sequences which sums up to a number less than 1000000. The rest is pretty straight forward, filter out the sums that aren’t prime numbers and find the longest sequence, and do this for all the primes we have generated to find the answer to the problem.

**Yan Cui**

I’m an **AWS Serverless Hero** and the author of **Production-Ready Serverless**. I have run production workload at scale in AWS for nearly 10 years and I have been an architect or principal engineer with a variety of industries ranging from banking, e-commerce, sports streaming to mobile gaming. I currently work as an independent consultant focused on AWS and serverless.

Further reading

Here is a complete list of all my posts on serverless and AWS Lambda. In the meantime, here are a few of my most popular blog posts.

- Lambda optimization tip – enable HTTP keep-alive
- You are thinking about serverless costs all wrong
- Many faced threats to Serverless security
- We can do better than percentile latencies
- I’m afraid you’re thinking about AWS Lambda cold starts all wrong
- Yubl’s road to Serverless
- AWS Lambda – should you have few monolithic functions or many single-purposed functions?
- AWS Lambda – compare coldstart time with different languages, memory and code sizes
- Guys, we’re doing pagination wrong