A natural number, N, that can be written as the sum and product of a given set of at least two natural numbers, {a1a2, ... , ak}
is called a product-sum number: N = a1 + a2 + ... + ak = a1 × a2 × ... × ak.

For example, 6 = 1 + 2 + 3 = 1 × 2 × 3.

For a given set of size, k, we shall call the smallest N with this property a minimal product-sum number. The minimal product-sum numbers for sets of size, k =
2, 3, 4, 5, and 6 are as follows.

k=2: 4 = 2 × 2 = 2 + 2

k=3: 6 = 1 × 2 × 3 = 1 + 2 + 3

k=4: 8 = 1 × 1 × 2 × 4 = 1 + 1 + 2 + 4

k=5: 8 = 1 × 1 × 2 × 2 × 2 = 1 + 1 + 2 + 2 + 2

k=6: 12 = 1 × 1 × 1 × 1 × 2 × 6 = 1 + 1 + 1 + 1 + 2 + 6

Hence for 2≤k≤6, the sum of all the minimal product-sum numbers is 4+6+8+12 = 30; note that 8 is only counted once in the sum.

In fact, as the complete set of minimal product-sum numbers for 2≤k≤12 is {4, 6, 8, 12, 15, 16}, the sum is 61.

What is the sum of all the minimal product-sum numbers for 2≤k≤12000?

n[k]表示minimal product-sum numbers for size=k

n[k]的上界为2*k,由于2*k总是能分解成2*k,然后2*k=k+2+(1)*(k-2)

显然n[k]的下界为k

对于一个数num   因式分解后因子个数为product   这些因子的和为sump

则须要加入的1的个数为num-sump,所以size k=num-sump+product

maxk = 12000
n=[2*maxk for i in range(maxk)] def getpsn(num,sump,product,start):
#print(num,' ',sump,' ',product)
k = num - sump + product
if k < maxk:
if num < n[k]:
n[k] = num
for i in range(start,maxk//num * 2): #控制num<=2*maxk
getpsn(num * i,sump + i,product + 1,i) getpsn(1,1,1,2)
ans=sum(set(n[2:]))
print(ans)

Project Euler:Problem 88 Product-sum numbers的更多相关文章

  1. Project Euler:Problem 61 Cyclical figurate numbers

    Triangle, square, pentagonal, hexagonal, heptagonal, and octagonal numbers are all figurate (polygon ...

  2. Project Euler:Problem 42 Coded triangle numbers

    The nth term of the sequence of triangle numbers is given by, tn = ½n(n+1); so the first ten triangl ...

  3. Project Euler:Problem 55 Lychrel numbers

    If we take 47, reverse and add, 47 + 74 = 121, which is palindromic. Not all numbers produce palindr ...

  4. Project Euler:Problem 87 Prime power triples

    The smallest number expressible as the sum of a prime square, prime cube, and prime fourth power is ...

  5. Project Euler: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 forme ...

  6. Project Euler:Problem 32 Pandigital products

    We shall say that an n-digit number is pandigital if it makes use of all the digits 1 to n exactly o ...

  7. Project Euler:Problem 76 Counting summations

    It is possible to write five as a sum in exactly six different ways: 4 + 1 3 + 2 3 + 1 + 1 2 + 2 + 1 ...

  8. Project Euler:Problem 34 Digit factorials

    145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145. Find the sum of all numbers which are ...

  9. Project Euler:Problem 89 Roman numerals

    For a number written in Roman numerals to be considered valid there are basic rules which must be fo ...

随机推荐

  1. 【POJ 2891】 Strange Way to Express Integers

    [题目链接] http://poj.org/problem?id=2891 [算法] exgcd [代码] #include <algorithm> #include <bitset ...

  2. python 编写的经纬度坐标转换类

    # -*- coding: utf-8 -*- # /** # * 各地图API坐标系统比较与转换; # * WGS84坐标系:即地球坐标系,国际上通用的坐标系.设备一般包含GPS芯片或者北斗芯片获取 ...

  3. BZOJ 3991 set维护dfs序

    思路: set按照dfn排序 两点之间的距离可以O(logn)算出来 加一个点-> now ans+=dis(pre,now)+dis(now,next)-dis(pre-next); 删一个点 ...

  4. 如何在C#中运行数学表达式字符串

    方法1:利用DataTable中的Compute方法 1 string expression = "1+2*3"; 2 DataTable eval = new DataTable ...

  5. 洛谷P1208 [USACO1.3]混合牛奶 Mixing Milk(贪心)

    题目描述 由于乳制品产业利润很低,所以降低原材料(牛奶)价格就变得十分重要.帮助Marry乳业找到最优的牛奶采购方案. Marry乳业从一些奶农手中采购牛奶,并且每一位奶农为乳制品加工企业提供的价格是 ...

  6. Intel VTune Amplifier XE 使用

    VTune <VTune 开发者手册> 1. 安装 1.1 软件安装 下载: (安装包下载地址) 安装: # 1.解压 tar -zxvf filename.tar.gz # 2.安装 c ...

  7. 剖析Promise内部结构,一步一步实现一个完整的、能通过所有Test case的Promise类

    本文写给有一定Promise使用经验的人,如果你还没有使用过Promise,这篇文章可能不适合你,建议先了解Promise的使用 Promise标准解读 1.只有一个then方法,没有catch,ra ...

  8. WordPress的wordfence插件的设置方法

  9. (转)webpack用法

    前言 webpack前端工程中扮演的角色越来越重要,它也是前端工程化很重要的一环.本文将和大家一起按照项目流程学习使用wbepack,妈妈再也不用担心我不会使用webpack,哪里不会看哪里.这是一个 ...

  10. List 常用方法解析

    1.Count属性 (获得List中元素数目) 2.Add( ) 在List中添加一个对象的公有方法 3.AddRange( ) 公有方法,在List尾部添加实现了ICollection接口的多个元素 ...