冪乗

  • $2^{31} - 1 = 2,147,483,647 \approx 2.1\times 10^9$
  • $2^{63} - 1 = 9,223,372,036,854,775,807 \approx 9.2 \times 10^{18}$
  • $3^{17} \approx 1.3 \times 10^8$
  • $3^{39} \approx 4.1 \times 10^{18}$

階乗，組合せ

  • $8! = 40,320$
  • $10! \approx 3.6 \times 10^6$
  • $20! \approx 2.4 \times 10^{18}$
$n$ $6$ $10$ $16$ $20$ $30$ $40$
$\binom{n}{n/2}$$20$$252$$12,870$$184,756$$1.5\times 10^{8}$ $1.4\times 10^{11}$

参考: Pythonでの計算方法

約数の個数

$n$ 以下の数の約数の個数の最大値．

$n$$10$ $10^5$ $10^8$ $10^9$ $10^{12}$ $10^{15}$
個数$4$ $128$ $768$ $1344$ $6720$ $26880$

参考: より詳しい表 ，高度合成数

1000までの素数表

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, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997

素数の分布

$\pi(x)$: $x$ 以下の素数の数

$x$ $10$$10^2$$10^3$$10^4$$10^5$$10^6$
$\pi(x)$ $4$$25$$168$$1229$$9592$$78498$

$\pi(x) \sim \frac{x}{\log x}$ である．

$x$ $10^6$$10^8$$10^{12}$$10^{16}$
$x / \log x$ $7.2\times10^4$$5.4\times10^6$$3.6\times10^{10}$$2.7\times10^{14}$