论坛 › 编程擂台 › 如何判断[根号(2^n)]的奇偶性

王守恩 发表于 6 天前

如何判断[根号(2^n)]的奇偶性

如何判断[根号(2^n)]的奇偶性？

Table], 2], {n, 98}]——标准。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0}
Table, 2], {n, 3}]——跟标准比,相同数位=2, a(1) = 2。
{1, 0, 1}
Table, 2], {n, 9}]——跟标准比,相同数位=8, a(2) = 8。
{1, 0, 0, 0, 1, 0, 1, 0, 1}
Table, 2], {n, 13}]——跟标准比,相同数位=12, a(3) = 12。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1}
Table, 2], {n, 20}]——跟标准比,相同数位=19, a(4) = 19。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1}
Table, 2], {n, 25}]——跟标准比,相同数位=24, a(5) = 24。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1}
Table, 2], {n, 31}]——跟标准比,相同数位=30, a(6) = 30。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1}
Table, 2], {n, 40}]——跟标准比,相同数位=39, a(7) = 39。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1}
Table, 2], {n, 43}]——跟标准比,相同数位=42, a(8) = 42。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1}
Table, 2], {n, 51}]——跟标准比,相同数位=50, a(9) = 50。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1}
Table, 2], {n, 59}]——跟标准比,相同数位=58, a(10) = 58。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1}
Table, 2], {n, 59}]——跟标准比,相同数位=58, a(11) = 58。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1}
Table, 2], {n, 59}]——跟标准比,相同数位=58, a(12) = 58。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1}
Table, 2], {n, 77}]——跟标准比,相同数位=76, a(13) = 76。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1}
Table, 2], {n, 77}]——跟标准比,相同数位=76, a(14) = 76。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1}
Table, 2], {n, 85}]——跟标准比,相同数位=84, a(15) = 84。
{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1}

得到一串数——2, 8, 12, 19, 24, 30, 39, 42, 50, 58, 58, 58, 76, 76, 84, 88, 104, 110, 113, 120, 129, 138, 144, 144, 144, 156, 166, 170, 186, 186,——有规律吗？
Sqrt(2) =   1.4, 1,4,   2,1,   3,5,   6,   2,3,   7,   3,   0,9,   5,   0,    4,    8,   8,   0,    1,   6,    8,   8,    7,   2,    4,   2,    0,   9,   69807856967187537694807317667973799073......

mathe 发表于 6 天前

https://oeis.org/A004539

王守恩 发表于 6 天前

mathe 发表于 2025-10-15 14:42
https://oeis.org/A004539
A004539——Expansion of sqrt(2) in base 2.——可能不是这串数。

{1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 0, 0,

Mod], 2],——我们这串数有个显著特征——不会有2个“1”连在一起。

{1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0}

王守恩 发表于 前天 09:05

是否存在无穷多正整数 n 满足 n 与 2^n 第一位数字相同?

a(1)=6,   2^6=64。
a(2)=10, 2^10=1024。
a(3)=14, 2^14=16384。
a(4)=17, 2^17=131072。
a(5)=21, 2^21=2097152。
a(6)=28, 2^28=268435456,。
a(7)=35, 2^35=34359738368。
a(8)=42, 2^42=4398046511104。
a(9)=59, 2^59=576460752303423488。

northwolves 发表于 前天 11:07

Select] == First &]

{6,10,14,17,21,28,35,42,59,76,93,100,103,107,110,113,117,120,123,127,130,133,137,140,143,147,150,153,157,160,163,167,170,173,177,180,183,187,190,193,196,204,207,214,217,224,227,234,237,244,247,254,257,264,267,277,287,297}

northwolves 发表于 前天 11:12

A100129
Numbers k such that 2^k starts with k.

6, 10, 1542, 77075, 113939, 1122772, 2455891300, 2830138178, 136387767490, 2111259099790, 3456955336468, 4653248164310, 10393297007134, 321249146279171, 972926121017616, 72780032758751764

查看完整版本: 如何判断[根号(2^n)]的奇偶性