## Project Euler个人解答(Problem 21~30)

2013年12月18日 发表评论 阅读评论

# Problem 21:Amicable numbers

Let d(n) be defined as the sum of proper divisors of n (numbers less than n which divide evenly into n).
If d(a) = b and d(b) = a, where a ≠ b, then a and b are an amicable pair and each of a and b are called amicable numbers.

For example, the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110; therefore d(220) = 284. The proper divisors of 284 are 1, 2, 4, 71 and 142; so d(284) = 220.

Evaluate the sum of all the amicable numbers under 10000.

d(n)定义为n 的所有真因子（小于 n 且能整除 n 的整数）之和。

### Code：Python

def d(n):
sum = 0;
for i in range(1,int(sqrt(n))+1):
if mod(n,i)==0:
sum += (i+n/i*(i!=n/i));
return sum-n

M = 10000;
sum = 0;
for i in range(1,M+1):
d1 = d(i);
if d1 != i and d(d1)==i:
sum += (i+d1);
print sum/2


# Problem 22:Names scores

Using names.txt (right click and ‘Save Link/Target As…’), a 46K text file containing over five-thousand first names, begin by sorting it into alphabetical order. Then working out the alphabetical value for each name, multiply this value by its alphabetical position in the list to obtain a name score.

For example, when the list is sorted into alphabetical order, COLIN, which is worth 3 + 15 + 12 + 9 + 14 = 53, is the 938th name in the list. So, COLIN would obtain a score of 938×53 = 49714.

What is the total of all the name scores in the file?

### Code：Mathematica

namefile = Import["ID22.\\txt"];
namelist = StringCases[namefile, "\"" ~~ (x : LetterCharacter ..) ~~ "\"" -> x];
namelist = Sort[namelist];
Total@Table[i*Plus @@ (Flatten[ToCharacterCode@Characters[namelist[[i]]]] -
64), {i, 1, Length[namelist]}]


# Problem 23:Non-abundant sums

A perfect number is a number for which the sum of its proper divisors is exactly equal to the number. For example, the sum of the proper divisors of 28 would be 1 + 2 + 4 + 7 + 14 = 28, which means that 28 is a perfect number.

A number n is called deficient if the sum of its proper divisors is less than n and it is called abundant if this sum exceeds n.

As 12 is the smallest abundant number, 1 + 2 + 3 + 4 + 6 = 16, the smallest number that can be written as the sum of two abundant numbers is 24. By mathematical analysis, it can be shown that all integers greater than 28123 can be written as the sum of two abundant numbers. However, this upper limit cannot be reduced any further by analysis even though it is known that the greatest number that cannot be expressed as the sum of two abundant numbers is less than this limit.

Find the sum of all the positive integers which cannot be written as the sum of two abundant numbers.

12是最小的过剩数，1 + 2 + 3 + 4 + 6 = 16。因此最小的能够写成两个过剩数之和的数字是24。经过分析，可以证明所有大于28123的数字都可以被写成两个过剩数之和。但是这个上界并不能被进一步缩小，即使我们知道最大的不能表示为两个过剩数之和的数字要比这个上界小。

### Code：Mathematica

(*判断是否是Non-abundant的函数*)
IsAbundantNum[n_] := Total@Divisors[n] - n > n;
(*计算所有Non-abundant数*)
AbundantNumList = Select[Range[28123], IsAbundantNum];
(*如果一个数和AbundantNumList的差与AbundantNumList之间交集为0，那么不可以分解成两个数的和*)
CanNotBeDevide[n_] :=
Length[Intersection[n - AbundantNumList, AbundantNumList]] ==  0
(*计算结果*)
Select[Range[28123], CanNotBeDevide] // Total



# Problem 24:Lexicographic permutations

A permutation is an ordered arrangement of objects. For example, 3124 is one possible permutation of the digits 1, 2, 3 and 4. If all of the permutations are listed numerically or alphabetically, we call it lexicographic order. The lexicographic permutations of 0, 1 and 2 are:

012 021 102 120 201 210

What is the millionth lexicographic permutation of the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9?

012 021 102 120 201 210

0, 1, 2, 3, 4, 5, 6, 7, 8，9的第100万个字典排列是什么？

Mathematica的话直接一行输出。
Python可以利用生成器很快的写一个出来，一直yield到需要的为止；

$$\dfrac{1000000}{9!}=2.75573$$

$$1000000-(9!\times 2)=274240$$

### Code：Python

def QPL(m_list):
if len(m_list) == 1:
yield m_list
for i in range(len(m_list)):
restlist = m_list[0:i]+m_list[i+1:]
for subres in QPL(restlist):
yield [m_list[i]]+subres;

c = 0;
for t in QPL(range(10)):
c += 1;
if c == 1000000:
print t;
break;


### Code：Mathematica

Permutations[Range[0, 9]][[1000000]]


# Problem 25:1000-digit Fibonacci number

The Fibonacci sequence is defined by the recurrence relation:
Fn = Fn-1 + Fn-2, where F1 = 1 and F2 = 1.
Hence the first 12 terms will be:
F1 = 1
F2 = 1
F3 = 2
F4 = 3
F5 = 5
F6 = 8
F7 = 13
F8 = 21
F9 = 34
F10 = 55
F11 = 89
F12 = 144
The 12th term, F12, is the first term to contain three digits.
What is the first term in the Fibonacci sequence to contain 1000 digits?

Fn = Fn-1 + Fn-2, where F1 = 1 and F2 = 1.

F1 = 1
F2 = 1
F3 = 2
F4 = 3
F5 = 5
F6 = 8
F7 = 13
F8 = 21
F9 = 34
F10 = 55
F11 = 89
F12 = 144

### Code：Mathematica

NestWhile[# + 1 &, 1, Length@IntegerDigits@Fibonacci[#] < 1000 &]


# Problem 26:Reciprocal cycles

A unit fraction contains 1 in the numerator. The decimal representation of the unit fractions with denominators 2 to 10 are given:

1/2 = 0.5
1/3 = 0.(3)
1/4 = 0.25
1/5 = 0.2
1/6 = 0.1(6)
1/7 = 0.(142857)
1/8 = 0.125
1/9 = 0.(1)
1/10 = 0.1
Where 0.1(6) means 0.166666..., and has a 1-digit recurring cycle. It can be seen that 1/7 has a 6-digit recurring cycle.

Find the value of d < 1000 for which 1/d contains the longest recurring cycle in its decimal fraction part.

1/2 = 0.5
1/3 = 0.(3)
1/4 = 0.25
1/5 = 0.2
1/6 = 0.1(6)
1/7 = 0.(142857)
1/8 = 0.125
1/9 = 0.(1)
1/10 = 0.1

### Code：Python

def clyLen(n):
cyl = [];
k = 10;
while([k,n] not in cyl):
cyl.append([k,n])
k = (k - int(k/n)*n)*10;
return len(cyl)-cyl.index([k,n])
maxlen = 0;
idx = 0;
for i in range(1,1001):
le = clyLen(i);
if le <= maxlen:
idx = i
maxlen = le
print idx


# Problem 27:Quadratic primes

Euler discovered the remarkable quadratic formula:

n2 + n + 41

It turns out that the formula will produce 40 primes for the consecutive values n = 0 to 39. However, when n = 40, 402 + 40 + 41 = 40(40 + 1) + 41 is divisible by 41, and certainly when n = 41, 412 + 41 + 41 is clearly divisible by 41.

The incredible formula n² -79n + 1601 was discovered, which produces 80 primes for the consecutive values n = 0 to 79. The product of the coefficients, -79 and 1601, is 126479.

Considering quadratics of the form:

n² + an + b, where |a| < 1000 and |b| < 1000

where |n| is the modulus/absolute value of n
e.g. |11| = 11 and |-4| = 4

Find the product of the coefficients, a and b, for the quadratic expression that produces the maximum number of primes for consecutive values of n, starting with n = 0.

n² + n + 41

n² + an + b, 其中|a| < 1000， |b| < 1000

|k²-k+41|<1000

### Code：Python

def isPrime(n):
if n <= 1:
return False
for i in xrange(2,int(n**0.5+1)):
if n%i == 0:
return False;
return True;

def GetPrimeLength(a,b):
cc = 0;
n = 0;
while(isPrime(n*n+a*n+b)):
n += 1;
cc += 1;
return cc

maxprimecoe = -1;
abset = [-1000,-1000];
for a in range(-1000,1001):
for b in range(-1000,1001):
curl = GetPrimeLength(a,b)
if curl < maxprimecoe:
maxprimecoe = curl;
abset = [a,b]
print maxprimecoe
print maxprimecoe,abset


# Problem 28:Number spiral diagonals

Starting with the number 1 and moving to the right in a clockwise direction a 5 by 5 spiral is formed as follows:

21 22 23 24 25
20  7  10
19  6  2  11
18  5  12
17 16 15 14 13

It can be verified that the sum of the numbers on the diagonals is 101.

What is the sum of the numbers on the diagonals in a 1001 by 1001 spiral formed in the same way?

21 22 23 24 25
20  7  10
19  6  2  11
18  5  12
17 16 15 14 13

1001×1001的螺旋中对角线上数字之和是多少？

2,2,2,2,4,4,4,4,6,6,6,6,8,8,8,8...

### Code：Python

Layer = 1001;
L = (Layer-1)/2;
sum = 1;
cur = 1;
step = 2;
for i in range(1,int(L)+1):
sum += cur*4+step*10
cur += step*4
step += 2
print sum


# Problem 29:Distinct powers

Consider all integer combinations of ab for 2 ≤ a ≤ 5 and 2 ≤ b ≤ 5:

22=4, 23=8, 24=16, 25=32
32=9, 33=27, 34=81, 35=243
42=16, 43=64, 44=256, 45=1024
52=25, 53=125, 54=625, 55=3125
If they are then placed in numerical order, with any repeats removed, we get the following sequence of 15 distinct terms:

4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125

How many distinct terms are in the sequence generated by ab for 2 ≤ a ≤ 100 and 2 ≤ b ≤ 100?

22=4, 23=8, 24=16, 25=32
32=9, 33=27, 34=81, 35=243
42=16, 43=64, 44=256, 45=1024
52=25, 53=125, 54=625, 55=3125

4, 8, 9, 16, 25, 27, 32, 64, 81, 125, 243, 256, 625, 1024, 3125

ab 在 2 ≤ a ≤ 100，2 ≤ b ≤ 100 下生成的序列中有多少个不同的项？

### Code：Python

sets = []
for a in range(2,101):
for b in range(2,101):
t = a**b;
if t not in sets:
sets.append(t)
print len(sets)


# Problem 30:Digit fifth powers

Surprisingly there are only three numbers that can be written as the sum of fourth powers of their digits:

1634 = 14 + 64 + 34 + 44
8208 = 84 + 24 + 04 + 84
9474 = 94 + 44 + 74 + 44
As 1 = 14 is not a sum it is not included.

The sum of these numbers is 1634 + 8208 + 9474 = 19316.

Find the sum of all the numbers that can be written as the sum of fifth powers of their digits.

1634 = 14 + 64 + 34 + 44
8208 = 84 + 24 + 04 + 84
9474 = 94 + 44 + 74 + 44
1 = 14没有被算作在内因为它不是一个和。

Plot[{10^x - 1, 9^5x}, {x, 1, 6}]


### Code：Mathematica

Select[Range[1, 400000], Plus @@ (Power[#, 5] & /@ IntegerDigits[#]) == # &]
Total[%] - 1


【完】

1. 2014年1月5日20:41 | #1

pe23：With[{a = Select[Range@28123, DivisorSigma[1, #] > 2 # &]}, Compile[{}, Block[{t = Range@28123}, Do[t = Complement[t, i + a], {i, a}]; t]][] ] // Tr // AbsoluteTiming pe30：Pick[#, # – Total[IntegerDigits[#]^5] & /@ #, 0] &@Range[2, 10^6] // Tr // AbsoluteTiming Sum[If[n == Total[IntegerDigits ^5], n, 0] , {n, 2, 10^6}] // AbsoluteTiming

• 2014年1月5日23:45 | #2

23题用do和Complement来实现这个想法好厉害。。。

• 2014年1月5日23:46 | #3

其实有时候我不是很敢用sum，因为感觉不是很好判断sum什么时候可以算得出来，什么时候算不出来。。

• 2014年1月6日12:35 | #4

Sum有些时候能自动编译，Pick可以充分利用向量化，Select要加速一般需要手动Compile

• 2014年1月5日23:48 | #5

看到30题想到一个问题像请教一下：不知大大有没有遇到过这种情况，就是对于&定义纯函数的时候，经常纯函数里面套纯函数，然后最里层的#就不知道应该是属于哪一层的了，这种时候我一般就改用Function函数来标记一下变量名，不知道有没有别的方法？【不知道我说清楚了没？

• 2014年1月6日13:12 | #6

还可以用With、Table等