发布时间:2019-09-27 07:09:28编辑:auto阅读(1790)
1、求100万以内的所有素数
质数又称素数,除了1和它本身以外不再有其他因数的数称为质数。
0和1都既不是质数也不是合数
为什么质数检验到N的开方就可以结束了?
因为所有N的除数都是以根号N为轴对称的
假如一个数N是合数,它有一个约数a,a×b=N,则a、b两个数中必有一个大于或等于根号N,一个小于或等于根号N;因此,只要小于或等于根号N的数(1除外)不能整除N,则N一定是素数.
所以判断一个数为质数与否的复杂度为O(根号N)
In [16]: num = int(input("Please a num: ")) #效率不高,打印出1000000以内的所有素数估计要几个小时 ...: for i in range(2, num): ...: for j in range(2, i): #时间复杂度为0(n**2) ...: if i % j == 0: ...: break ...: else: ...: print(i) ...: Please a num: 10 2 3 5 7 In [18]: num = int(input("Please a num: ")) ...: for i in range(2, num): ...: for j in range(2, i): ...: if i % j == 0: ...: break ...: else: ...: print(i) ...: Please a num: 100 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
正确做法:
In [20]: num = int(input("Please a num: ")) ...: c = 0 ...: for i in range(2, num): ...: for j in range(2, int(i**0.5)+1): ...: if i % j == 0: ...: break ...: else: ...: c += 1 ...: print(i) ...: print("The total num is: ", c) ...: Please a num: 10 2 3 5 7 The total num is: 4 In [21]: num = int(input("Please a num: ")) ...: c = 0 ...: for i in range(2, num): ...: for j in range(2, int(i**0.5)+1): ...: if i % j == 0: ...: break ...: else: ...: c += 1 ...: print(i) ...: print("The total num is: ", c) ...: Please a num: 100 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 The total num is: 25 In [22]: num = int(input("Please a num: ")) ...: c = 0 ...: for i in range(2, num): ...: for j in range(2, int(i**0.5)+1): ...: if i % j == 0: ...: break ...: else: ...: c += 1 ...: # print(i) ...: print("The total num is: ", c) ...: Please a num: 1000000 The total num is: 78498
2、求任意整数的几何级数
无穷级数中,几何级数又称为等比级数。
几何级数(即等比级数)的和为:当︱q︱<1时a+aq+aq^2+……+aq^n+……=a/(1-q)
In [44]: for n in range(11): ...: sum += 3 * 2**n ...: print(sum) ...: 3 9 21 45 93 189 381 765 1533 3069 6141 In [45]
3、求菲波那切数列的第101位
斐波那契数列指的是这样一个数列 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233,377,610,987,1597,2584,4181,6765,10946,17711,28657,46368........
这个数列从第3项开始,每一项都等于前两项之和
In [52]: for i in range(0, 102): ...: if i == 0: ...: y = 1 ...: elif i == 1: ...: x = 1 ...: y = 1 ...: else: ...: tmp = y ...: y = x + y ...: x = tmp ...: print(y) ...: 927372692193078999176 In [21]: a=0 In [22]: b=1 In [23]: i=0 In [24]: while i <= 101: ...: print(b) ...: a, b, i= b, a+b, i+1 ...: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040 1346269 2178309 3524578 5702887 9227465 14930352 24157817 39088169 63245986 102334155 165580141 267914296 433494437 701408733 1134903170 1836311903 2971215073 4807526976 7778742049 12586269025 20365011074 32951280099 53316291173 86267571272 139583862445 225851433717 365435296162 591286729879 956722026041 1548008755920 2504730781961 4052739537881 6557470319842 10610209857723 17167680177565 27777890035288 44945570212853 72723460248141 117669030460994 190392490709135 308061521170129 498454011879264 806515533049393 1304969544928657 2111485077978050 3416454622906707 5527939700884757 8944394323791464 14472334024676221 23416728348467685 37889062373143906 61305790721611591 99194853094755497 160500643816367088 259695496911122585 420196140727489673 679891637638612258 1100087778366101931 1779979416004714189 2880067194370816120 4660046610375530309 7540113804746346429 12200160415121876738 19740274219868223167 31940434634990099905 51680708854858323072 83621143489848422977 135301852344706746049 218922995834555169026 354224848179261915075 573147844013817084101 927372692193078999176 In [25]:
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