➗ 数学工具Math （第 6 页）(page 6)

整数相乘Multiply Integers

把两个或多个整数相乘，支持任意大小。本地处理。可设置「千位逗号」。Multiply two or more integers of any size. Processed locally. Settings: Thousands commas.

🧹

去除整数中的千位逗号Remove Commas from an Integer

去掉整数写法里的千位逗号（1,234,567 → 1234567），可批量处理，保留负号。本地处理。Strip thousands commas from integer notation (1,234,567 -> 1234567), in bulk, keeping signs. Processed locally.

🎯

从整数列表中随机抽取一个或多个（可设种子复现，可要求不重复）。适合抽签和抽样，本地处理。Pick one or more integers at random from a list (optional seed for reproducibility; optional unique picks). Handy for draws and sampling. Processed locally.

🔀

替换整数中的数字Replace Integer Digits

把每个整数里的指定数字替换成另一个（例如把所有 0 换成 9），保留符号。本地处理。Replace one digit with another in each integer (e.g. turn every 0 into 9), keeping the sign. Processed locally.

🔁

反转整数各位数字Reverse Integer Digits

把每个整数的数字左右颠倒（120 → 021），可选择去掉结果前导零。本地处理。Reverse the digits of each integer (120 -> 021); optionally drop leading zeros in the result. Processed locally.

➕

把整数改写成若干数之和Rewrite Integer as Sum

把每个正整数改写成若干正整数之和：按固定加数（如全是 1）、按十进制位权展开，或随机拆分（可设种子）。本地处理。Rewrite each positive integer as a sum of positive integers: fixed addends (e.g. all 1s), decimal place-value expansion, or a random split (optional seed). Processed locally.

✖️

把整数改写成若干数之积Rewrite Integer as Product

把每个大于 1 的整数改写成因数乘积：质因数分解，或拆成若干大于 1 的因数（随机，可设种子）。本地处理。Rewrite each integer greater than 1 as a product of factors: prime factorization, or a random factor split (optional seed). Processed locally.

🎲

打乱整数各位数字Shuffle Integer Digits

随机打乱每个整数的各位数字（可设种子），保留符号。本地处理。可设置「种子（留空 = 真随机）」、「去掉前导零」。Randomly shuffle the digits of each integer (optional seed), keeping the sign. Processed locally. Settings: Seed (blank = truly random), Drop leading zeros.

🎲

打乱整数列表顺序Shuffle Integers

随机打乱整数列表的顺序（可设种子复现）。本地处理。可设置「种子（留空 = 真随机）」、「分隔」、「千位逗号」。Randomly shuffle the order of a list of integers (optional seed). Processed locally. Settings: Seed (blank = truly random), Separator, Thousands commas.

🔢

排序整数各位数字Sort Integer Digits

把每个整数的各位数字按升序或降序排列，保留符号。本地处理。可设置「顺序」、「去掉前导零」。Sort the digits of each integer ascending or descending, keeping the sign. Processed locally. Settings: Order, Drop leading zeros.

↕️

排序整数列表Sort Integers

把整数列表按数值升序或降序排序。本地处理。可设置「顺序」、「分隔」、「去重」、「千位逗号」。Sort a list of integers by numeric value, ascending or descending. Processed locally. Settings: Order, Separator, Remove duplicates, Thousands commas.

🔺

生成整数三元组（勾股等）Generate Integer Triples

生成整数三元组：勾股数（a^2+b^2=c^2）、连续三整数，或和为 S 的随机三元组（可设种子）。本地处理。Generate integer triples: Pythagorean triples, three consecutive integers, or random triples that sum to S (optional seed). Processed locally.

🪞

生成对称整数（回文数）Generate Symmetric Integers

生成回文整数：指定位数与个数，可设种子。本地处理。可设置「个数」、「位数」、「种子（留空 = 真随机）」。Generate palindromic integers of a chosen digit length and count; optional seed. Processed locally. Settings: How many, Digits, Seed (blank = truly random).

生成冯·诺依曼序数Generate von Neumann Ordinals

生成冯·诺依曼自然数（序数）的集合写法：0=∅, 1={∅}, 2={∅,{∅}}, …。可指定从 0 到 N。本地处理。Generate von Neumann naturals (ordinals) in set notation: 0 = empty set, 1 = set containing empty, 2 = set of those, and so on, from 0 through N. Processed locally.

生成策梅洛序数Generate Zermelo Ordinals

生成策梅洛自然数的集合写法：0=∅, 1={∅}, 2={{∅}}, 3={{{∅}}}, …。可指定从 0 到 N。本地处理。Generate Zermelo naturals in set notation: 0 = empty set, 1 = singleton of empty, 2 = singleton of that, and so on, from 0 through N. Processed locally.

✂

拆分整数（按位数 / 块）Split an Integer

把整数按固定位数或固定块数切开（例如 123456 按 2 位 → 12 34 56），可从左或从右切，保留符号。适合分组显示和教学，本地处理。Split an integer into fixed-size digit chunks or a fixed number of parts (e.g. 123456 by pairs -> 12 34 56), from the left or right, keeping the sign. Handy for grouped display and teaching. Processed locally.

✨

整数写成花体 / 数学字体Write Integers in Fancy Font

把整数的数字换成 Unicode 数学字母数字（粗体、无体、双线、无衬线粗体、等宽等），方便发帖和标题。本地处理。Rewrite integer digits in Unicode mathematical alphanumeric forms (bold, italic, double-struck, sans-bold, monospace, etc.) for posts and titles. Processed locally.

➗

找出全部因数Find All Divisors

列出一个整数的全部正因数（含 1 和它本身），可按行或逗号输出，并显示因数个数与之和。本地处理。List every positive divisor of an integer (including 1 and itself), as lines or commas, with the count and sum. Processed locally.

📈

生成盈数（丰数）Generate Abundant Numbers

生成真因数之和大于自身的盈数（丰数）。可设个数与起点，本地处理。可设置「个数」、「起点」。Generate abundant numbers (proper-divisor sum greater than the number). Set how many and where to start. Processed locally. Settings: How many, Start.

≈

生成几乎完全数Generate Almost Perfect Numbers

生成几乎完全数（真因数之和 = n−1）。已知的几乎完全数都是 2 的幂；本工具按此列出。本地处理。Generate almost perfect numbers (proper-divisor sum = n−1). All known examples are powers of two; this tool lists those. Processed locally.

📉

生成亏数Generate Deficient Numbers

生成真因数之和小于自身的亏数。可设个数与起点，本地处理。可设置「个数」、「起点」。Generate deficient numbers (proper-divisor sum less than the number). Set how many and where to start. Processed locally. Settings: How many, Start.

🌳

生成 Baum–Sweet 序列Generate Baum–Sweet Sequence

生成 Baum–Sweet 序列：若 n 的二进制表示不含奇数长度的连续 0 块则为 1，否则为 0。本地处理。Generate the Baum–Sweet sequence: 1 if the binary of n has no odd-length run of zeros, else 0. Processed locally.

𝑒

生成自然常数 e 的位数Generate e Digits

用级数 Σ 1/n! 和大整数精确生成 e 的小数点后 1–5000 位。本地处理。可设置「小数位数」。Generate 1–5000 decimal places of e = Σ 1/n! with exact big integers. Processed locally. Settings: Decimal places.

2️⃣

生成偶数Generate Even Numbers

生成一段偶数：可设个数、起点与步长（默认 2）。本地处理。可设置「个数」、「起点」、「步长」。Generate even numbers: set how many, the start value, and step (default 2). Processed locally. Settings: How many, Start, Step.

𝐹

生成斐波那契质数Generate Fibonacci Primes

列出既是斐波那契数又是质数的项（按斐波那契下标）。可设要找多少个，本地用 Baillie–PSW 判断（大整数）。List Fibonacci numbers that are prime (by Fibonacci index). Set how many to find; primality uses the Baillie–PSW test locally.

𝑾

生成斐波那契词Generate Fibonacci Words

生成斐波那契词：S0=0，S1=01（或 a/b 字母），S_n = S_{n-1}+S_{n-2}。可输出某一项或前几项，本地处理。Generate Fibonacci words: S0=0, S1=01 (or a/b letters), S_n = S_{n-1}+S_{n-2}. One term or the first few. Processed locally.

🔢

生成 Gijswijts 序列Generate Gijswijts Sequence

生成 Gijswijt 序列（OEIS A090822）：a(1)=1，之后每一项等于“目前序列末尾最多由同一段连续重复了几次”。1 1 2 1 1 2 2 2 3 1 1 2…，本地处理。Generate Gijswijt's sequence (OEIS A090822): a(1)=1, then each term is the largest k such that the sequence so far ends with some block repeated k times in a row. 1 1 2 1 1 2 2 2 3 1 1 2… Processed locally.

𝕀

生成单位矩阵Generate Identity Matrix

生成 n×n 单位矩阵（对角线为 1，其余为 0），可调分隔符。本地处理。可设置「阶数 n」、「分隔」。Generate an n×n identity matrix (1 on the diagonal, 0 elsewhere); choose the separator. Processed locally. Settings: Order n, Separator.

🌀

生成 Kolakoski 序列Generate Kolakoski Sequence

生成 Kolakoski 序列（自描述的 1/2 游程序列）：1,2,2,1,1,2,1,2,2,1,…。本地处理。Generate the Kolakoski sequence (self-describing 1/2 run-length sequence): 1,2,2,1,1,2,1,2,2,1,… Processed locally.

👀

生成外观数列（读数序列）Generate Look-and-Say Sequence

生成外观数列（look-and-say）：从种子出发，每次用「几个几」描述上一项。可设项数与种子，本地处理。Generate the look-and-say sequence: each term describes the previous as runs of digits. Set terms and seed. Processed locally.

𝑳

生成卢卡斯数Generate Lucas Numbers

生成卢卡斯数列：L0=2，L1=1，之后每项为前两项之和。可设项数与起点下标，本地处理。Generate Lucas numbers: L0=2, L1=1, then each term is the sum of the previous two. Set count and start index. Processed locally.

𝑳′

生成卢卡斯质数Generate Lucas Primes

列出既是卢卡斯数又是质数的项。可设要找多少个，本地用 Baillie–PSW 判断（大整数）。可设置「个数」。List Lucas numbers that are prime. Set how many to find; primality uses the Baillie–PSW test locally. Settings: How many.

⬚

生成 Moser–de Bruijn 序列Generate Moser–de Bruijn Sequence

生成 Moser–de Bruijn 序列：把 n 的二进制位当作四进制数字得到的数（只含数字 0、1）。本地处理。Generate the Moser–de Bruijn sequence: interpret the binary digits of n as base-4 digits (only 0 and 1). Processed locally.

−𝐹

生成负斐波那契数Generate NegaFibonacci Numbers

生成负下标斐波那契数：F_{-n}=(-1)^{n+1} F_n。可设个数与起始负下标，本地处理。Generate negative-index Fibonacci numbers: F_{-n}=(-1)^{n+1} F_n. Set count and start index. Processed locally.

−𝑳

生成负卢卡斯数Generate NegaLucas Numbers

生成负下标卢卡斯数：L_{-n}=(-1)^n L_n。可设个数与起点，本地处理。可设置「项数」、「起点下标（负）」。Generate negative-index Lucas numbers: L_{-n}=(-1)^n L_n. Set count and start. Processed locally. Settings: Terms, Start index (negative).

1️⃣

生成奇数Generate Odd Numbers

生成一段奇数：可设个数、起点与步长（默认 2）。本地处理。可设置「个数」、「起点」、「步长」。Generate odd numbers: set how many, start, and step (default 2). Processed locally. Settings: How many, Start, Step.

📄

生成折纸序列Generate Paperfolding Sequence

生成规则折纸（龙曲线）序列 OEIS A014577：反复向同一方向对折纸条，展开后的折痕方向，1 表示谷折、0 表示峰折。本地处理。Generate the regular paperfolding (dragon curve) sequence, OEIS A014577: fold a strip in half repeatedly in the same direction and read the creases after unfolding, 1 for a valley fold and 0 for a mountain fold. Processed locally.

🎲

生成随机矩阵Generate Random Matrix

生成随机整数矩阵：可设行列、数值范围与种子。本地处理。可设置「行数」、「列数」、「下限」、「上限」、「种子（可选）」。Generate a random integer matrix: set rows, columns, value range and seed. Processed locally. Settings: Rows, Columns, Min, Max, Seed (optional).

生成 Rudin–Shapiro 序列Generate Rudin–Shapiro Sequence

生成 Rudin–Shapiro 序列：±1 值，与二进制中相邻 11 的个数奇偶相关。本地处理。可设置「项数」、「每行一个」。Generate the Rudin–Shapiro sequence of ±1 from the parity of adjacent 11-bits in n. Processed locally. Settings: Terms, One per line.

生成 Thue–Morse 序列Generate Thue–Morse Sequence

生成 Thue–Morse 序列：n 的二进制中 1 的个数的奇偶（0/1）。本地处理。可设置「项数」、「每行一个」。Generate the Thue–Morse sequence: parity of the number of 1-bits in n (0/1). Processed locally. Settings: Terms, One per line.

🔤

生成 Tribonacci 词Generate Tribonacci Words

生成 Tribonacci 词（OEIS A080843，字母 a、b、c 写作 0、1、2）。下标从 0 开始：S0 = 0，S1 = 01，S2 = 0102，之后 Sn = Sn−1 Sn−2 Sn−3（把前三项依次拼接）；Sn 也等于从 0 出发做 n 次替换 0→01、1→02、2→0。本地处理。Generate Tribonacci words (OEIS A080843, with the letters a, b, c written as 0, 1, 2). Indices start at 0: S0 = 0, S1 = 01, S2 = 0102, then Sn = Sn−1 Sn−2 Sn−3 (the previous three joined in order); Sn is also what n substitutions 0→01, 1→02, 2→0 make from 0. Processed locally.

⁻¹

矩阵求逆Invert Matrix

求方阵的逆：把输入的整数、小数或分数当作精确有理数，用高斯–约当消元精确求逆，再按需显示为小数或分数。本地处理。Invert a square matrix: integers, decimals and fractions are read as exact rationals and inverted exactly with Gauss–Jordan elimination, then shown as decimals or fractions. Processed locally.

矩阵乘法Matrix Multiply

计算两个矩阵的乘积 A×B，整数、小数和分数都精确计算。在上方输入框写 A，空一行再写 B；或者把 B 写进「矩阵 B」（用分号分行）。本地处理。Multiply two matrices, A×B, with integers, decimals and fractions computed exactly. Type A in the box above, then a blank line and B; or put B in the Matrix B field (rows separated by semicolons). Processed locally.

🌿