The Project Euler thread

[dag12]

Re: THE project euler thread

Sounds like you could just write a script for euler's totient function and then run a for loop or something >.>
Though I'm sure there's a more elegant way of doing it.

Also, any way to check answers without registering?

[Reincarnate]

Re: THE project euler thread

Sounds like you could just write a script for euler's totient function and then run a for loop or something >.>
Though I'm sure there's a more elegant way of doing it.

Also, any way to check answers without registering?

tried this :<

and no you have to register

[stargroup100]

Re: THE project euler thread

UGH

okay this problem is seriously impossible tackling it the way I'm doing it unless there's some beautiful optimization trick

this is not working

[Reincarnate]

Re: THE project euler thread

getting close but still off

the way I am tackling it is actually very similar to what you posed above (raising primes to powers) but figuring out where to fix things is tricky

For Co(10):
sorted mainlist [1, 5, 7, 8, 9]
sum of mainlist 30
primes [3, 5, 7]

this is fine

For Co(30):
sorted mainlist [1, 11, 13, 17, 19, 23, 25, 27, 28, 29]
sum of mainlist 193
primes [3, 5, 7, 11, 13, 17, 19, 23, 29]

this is correct too

But for Co(100):
sorted mainlist [1, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 97, 98]
sum of mainlist 1248
primes [3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

this is incorrect

[iironiic]

Re: THE project euler thread

Should 99 be in your sorted mainlist for Co(100)?

I'm thinking about an approach right now, but I haven't come up with anything yet.

[Reincarnate]

Re: THE project euler thread

Should 99 be in your sorted mainlist for Co(100)?

I'm thinking about an approach right now, but I haven't come up with anything yet.

sorted mainlist [1, 7, 11, 13, 16, 17, 19, 23, 25, 27, 29]
sum of mainlist 188
primes [2, 3, 5, 7, 11, 13, 17, 19, 23, 29]

This is what I get for Co(30) without that optimization and going for power-raising alone. Obviously I need to toss 28 into this beast, but in doing so I must remove 7 and 16.

EDIT: What I do now, in general is create the "naive prime-raised" solution and then add numbers to the solution and remove all coprime elements, then check to see if that sum is better than what I had before. If so, update solution. If not, kept what I had before and try a new number.

[Reincarnate]

Re: THE project euler thread

Hm somehow it's one off, wtf

sorted mainlist [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97, 99]
sum of mainlist 1355
primes [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]

[LongGone]

Re: THE project euler thread

Problem 355:

However, it is possible that a term has the form (p^a)(q^b), where p and q are primes, if (p^a)(q^b) > p^c + q^d. (where a,b,c,d are the highest possible powers such that each term is less than n)

Could this be extended to cover cases of (p^a)(q^b)(r^c) > p^d + q^e + r^f etc. ? I can't think of an example though. The huge annoyance here is that the list of numbers are exhaustive, everytime you form a set of p,q there could be another set which may use them more efficiently instead (pointing out the obvious though)

[stargroup100]

Re: THE project euler thread

I have a conjecture that a term of the form (p^a)(q^b)(r^c) is never found in a maximal sum subset, but I'm not sure if I can prove it. If this is true it would at least make the problem significantly easier. I forgot to mention this earlier.

[Reincarnate]

Re: THE project euler thread

can anyone find the problem in my earlier list? should equal 1356, not 1355 (according to the problem description)

[stargroup100]

Re: THE project euler thread

lurker had the same problem

I'm trying to find the error as well

[LongGone]

Re: THE project euler thread

can anyone find the problem in my earlier list? should equal 1356, not 1355 (according to the problem description)

Since you are one off from the actual answer (changing from odd to even), it implies that one set of (p^a)(q^b) should be degrouped, or grouped (e.g. 5*19=95(odd), or 5^a+19^b=even). Swapping pairs of number products will not change the odd/even parity
The main concern are the primes <10 in this case (because obviously you can't multiply two primes >10 that will result in <100 product)
So the usable primes are 2,3,5,7
Since you used up all 4 and ended up with 1355; If 1356 was not a mistake, that means that you have to degroup one of those primes and leave one as a natural power of itself (which is counterintuitive)
...So either the 13 people who solved didnt bother checking Co(100), or I'm missing something rofl
(or a rare case of a possible product of three primes, which idk if its plausible)

For a possible general approach, to try to maximise sum, we can start by getting rid of all the small primes
So e.g. for Co(100), start with 11,13,17,19. With your current solution we got: 99 (3,11), 95 (5,19), and 92 (2,23), 91 (7,13). Could there be a possible solution using 17 in place of 23?

[stargroup100]

Re: THE project euler thread

edit2: ok I was being stupid

EDIT: Rubix, your answer might be wrong but it looks pretty close :D

[Reincarnate]

Re: THE project euler thread

A step-by-step runthrough of the Co(100) process:

initial naive list from prime-raising alone: [1, 11, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97] with sum 1263

adding 99 to the list and removing non-coprimes to test [1, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 97, 99]
Updating list because biggestsum 1263 is less than 1270
adding 98 to the list and removing non-coprimes to test [1, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 98, 99]
adding 96 to the list and removing non-coprimes to test [1, 13, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 96, 97]
adding 95 to the list and removing non-coprimes to test [1, 13, 17, 23, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 95, 97, 99]
Updating list because biggestsum 1270 is less than 1321
adding 94 to the list and removing non-coprimes to test [1, 13, 17, 23, 29, 31, 37, 41, 43, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 94, 95, 97, 99]
adding 93 to the list and removing non-coprimes to test [1, 13, 17, 23, 29, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 93, 95, 97]
adding 92 to the list and removing non-coprimes to test [1, 13, 17, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97, 99]
Updating list because biggestsum 1321 is less than 1326
adding 91 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97, 99]
Updating list because biggestsum 1326 is less than 1355
adding 90 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 90, 91, 97]
adding 88 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 88, 89, 91, 95, 97]
adding 87 to the list and removing non-coprimes to test [1, 17, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 87, 89, 91, 92, 95, 97]
adding 86 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 86, 89, 91, 95, 97, 99]
adding 85 to the list and removing non-coprimes to test [1, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 85, 89, 91, 92, 97, 99]
adding 84 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 84, 89, 95, 97]
adding 82 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 43, 47, 53, 59, 61, 67, 71, 73, 79, 82, 83, 89, 91, 95, 97, 99]
adding 81 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 92, 95, 97]
adding 80 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 80, 83, 89, 91, 97, 99]
adding 78 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 78, 79, 83, 89, 95, 97]
adding 77 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 77, 79, 83, 89, 92, 95, 97]
adding 76 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 76, 79, 83, 89, 91, 97, 99]
adding 75 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 75, 79, 83, 89, 91, 92, 97]
adding 74 to the list and removing non-coprimes to test [1, 17, 29, 31, 41, 43, 47, 53, 59, 61, 67, 71, 73, 74, 79, 83, 89, 91, 95, 97, 99]
adding 72 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 72, 73, 79, 83, 89, 91, 95, 97]
adding 70 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 70, 71, 73, 79, 83, 89, 97, 99]
adding 69 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 69, 71, 73, 79, 83, 89, 91, 95, 97]
adding 68 to the list and removing non-coprimes to test [1, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 68, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 66 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 66, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 65 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 65, 67, 71, 73, 79, 83, 89, 92, 97, 99]
adding 64 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 64, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 63 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 63, 67, 71, 73, 79, 83, 89, 92, 95, 97]
adding 62 to the list and removing non-coprimes to test [1, 17, 29, 37, 41, 43, 47, 53, 59, 61, 62, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 60 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 60, 61, 67, 71, 73, 79, 83, 89, 91, 97]
adding 58 to the list and removing non-coprimes to test [1, 17, 31, 37, 41, 43, 47, 53, 58, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 57 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 57, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97]
adding 56 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 56, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 55 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 55, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97]
adding 54 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 53, 54, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 52 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 52, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 51 to the list and removing non-coprimes to test [1, 29, 31, 37, 41, 43, 47, 51, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 50 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 50, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 49 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97, 99]
adding 48 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 47, 48, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 46 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 46, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 45 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 45, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97]
adding 44 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 43, 44, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 42 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 41, 42, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97]
adding 40 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 40, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 39 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 39, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97]
adding 38 to the list and removing non-coprimes to test [1, 17, 29, 31, 37, 38, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 36 to the list and removing non-coprimes to test [1, 17, 29, 31, 36, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 35 to the list and removing non-coprimes to test [1, 17, 29, 31, 35, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 97, 99]
adding 34 to the list and removing non-coprimes to test [1, 29, 31, 34, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 33 to the list and removing non-coprimes to test [1, 17, 29, 31, 33, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 32 to the list and removing non-coprimes to test [1, 17, 29, 31, 32, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 30 to the list and removing non-coprimes to test [1, 17, 29, 30, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97]
adding 28 to the list and removing non-coprimes to test [1, 17, 28, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 27 to the list and removing non-coprimes to test [1, 17, 27, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 26 to the list and removing non-coprimes to test [1, 17, 26, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 25 to the list and removing non-coprimes to test [1, 17, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97, 99]
adding 24 to the list and removing non-coprimes to test [1, 17, 24, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 23 to the list and removing non-coprimes to test [1, 17, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 22 to the list and removing non-coprimes to test [1, 17, 22, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 21 to the list and removing non-coprimes to test [1, 17, 21, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97]
adding 20 to the list and removing non-coprimes to test [1, 17, 20, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 19 to the list and removing non-coprimes to test [1, 17, 19, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97, 99]
adding 18 to the list and removing non-coprimes to test [1, 17, 18, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 16 to the list and removing non-coprimes to test [1, 16, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 15 to the list and removing non-coprimes to test [1, 15, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97]
adding 14 to the list and removing non-coprimes to test [1, 14, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 95, 97, 99]
adding 13 to the list and removing non-coprimes to test [1, 13, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97, 99]
adding 12 to the list and removing non-coprimes to test [1, 12, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 11 to the list and removing non-coprimes to test [1, 11, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 10 to the list and removing non-coprimes to test [1, 10, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 97, 99]
adding 9 to the list and removing non-coprimes to test [1, 9, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 8 to the list and removing non-coprimes to test [1, 8, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 7 to the list and removing non-coprimes to test [1, 7, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 92, 95, 97, 99]
adding 6 to the list and removing non-coprimes to test [1, 6, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97]
adding 5 to the list and removing non-coprimes to test [1, 5, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 97, 99]
adding 4 to the list and removing non-coprimes to test [1, 4, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]
adding 3 to the list and removing non-coprimes to test [1, 3, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97]
adding 2 to the list and removing non-coprimes to test [1, 2, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 95, 97, 99]

sorted, final mainlist [1, 17, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 91, 92, 95, 97, 99]
sum of mainlist 1355

[stargroup100]

Re: THE project euler thread

Your solution is too linear. You should take into consideration the possible case that immediately adding the largest possible number won't optimize the overall total.

It's possible there is a different combination of the primes that rounds out to a better sum. I recommend this method:

- Separate primes into 3 categories: between 2 and sqrt(n), between sqrt(n) and n/2, and the others
- Add up all of the primes in the last category. These don't change.
- The terms in the second category have the most room for possible optimizations, and therefore should try to be optimized first. They obviously cannot be paired with a prime from its own category, so pair it with a prime from the first category. Try every combination here.
- The last possible case is that 2 primes from the first category might be paired together. Consider this generalization last.

Do it this way and see if you can get to 1356.


NOTE: I'm not sure if the last step is necessary, I'm trying to see if I can prove whether or not that last step is needed.