Interesting Numbers

-1 (the square of imaginary unit i)

0

0.207 879 576 350 761 ... (the approximate value of i^i)

0.301 029 995 663 981 ... (the approximate value of logarithm of 2 to base 10)

0.318 309 886 183 790 ... (the approximate value of π^-1)

0.367 879 441 171 442 ...  (the approximate value of e^-1)

0.577 215 664 901 532 ...  (the approximate value of Euler's constant γ)

0.693 147 180 559 945 ... (the approximate value of log 2)

1 (the unit)

1.259 921 049 894 873 ... (the approximate value of the cube root of 2)

1.414 213 562 373 095 ... (the approximate value of the square root of 2)

1.618 033 988 749 894 ... (the approximate value of the golden ratio)

1.732 050 807 568 877 ... (the approximate value of the square root of 3)

1.772 453 850 905 516 ... (the approximate value of the square root of π)

2 (the only even prime number)

2.302 585 092 994 045 ... (the approximate value of the natural logarithm of 10)

2.718 281 828 459 045 ... (the approximate value of the base of natural logarithm e)

3,4,5 (the smallest numbers satisfying Pythagoras' theorem)

4 (the first composite number, the first square of a prime number)

3.141 592 653 589 793 ... (the approximate value of π)

5 (the smallest number written as the sum of two squares)

6 = 1+2+3 (the smallest perfect number - number expressible as sum of its divisors)

8 (the only cube that is one less than a square, also the base of the octonary system)

10 (the base of the counting system)

15 (the first product of 2 odd primes)

22.459 157 718 361 045 ... (the approximate value of π^e)

23.140 692 632 779 269 ... (the approximate value of e^π)

42 (the magic constant of the smallest magic cube composed with the numbers 1 to 27)

220 & 284 (the smallest pair of amicable numbers - two different numbers related in such a way that the sum of the proper divisors of each is equal to the other number)

60 (the base of the sexadecimal system of counting; the interior angle of an equilateral triangle)

70 (the smallest weird number - a natural number that is abundant without being the sum of any set of its own divisors)

90 (the number of degrees in the right angle)

101 (the smallest palindromic prime - a prime number that is also a palindromic number)

145 = 1! + 4! + 5! (the smallest factoring number - numbers which are sum of the factorials of their own digits)

153 = 1³ + 5³ + 3³ (the smallest narcissistic number - n-digit numbers that are equal to the sum of the nth powers of their digits)

180 (the number of degrees in a half-circle)

360 (the number of degrees in the circle)

1729 = 1³ + 12³ = 9³ + 10³ (the smallest number expressible as a sum of two cubes)

12,946 → 14,288 → 15,742 → 14,536 → 14,264 → 12,946→... (5-cycle social numbers - numbers whose aliquot sums form a cyclic sequence that begins and ends with the same number)

Resources:
David Wells (1986) "The Penguin Dictionary of Curious and Interesting Numbers"