## Δευτέρα, 13 Μαρτίου 2017

### Οι πρώτοι 200 αριθμοί...

Τα παρακάτω τα βρήκα στη σελίδα του "I love Mathematics" στο Facebook. Πραγματικά εκπληκτικό και αν και είναι στα αγγλικά αξίζει να του ρίξετε μια ματιά...
Μερικά ενδιαφέροντα στοιχεία για τους πρώτους 200 αριθμούς...

1 is the multiplicative identity .
2 is the only even prime.
3 is the number of spatial dimensions we live in.
4 is the smallest number of colors sufficient to color all planar maps.
5 is the number of Platonic solids .
6 is the smallest perfect number.
7 is the smallest number of sides of a regular polygon that is not constructible by straightedge and compass.
8 is the largest cube in the Fibonacci sequence
9 is the maximum number of cubes that are needed to sum to any positive integer .
10 is the base of our number system.
11 is the largest known multiplicative persistence .
12 is the smallest abundant number.
13 is the number of Archimedean solids .
14 is the smallest even number n with no solutions to φ(m) = n.
15 is the smallest composite number n with the property that there is only one group of order n.
16 is the only number of the form x y = y x with x and y being different integers .
17 is the number of wallpaper groups .
18 is the only positive number that is twice the sum of its digits.
19 is the maximum number of 4 th powers needed to sum to any number.
20 is the number of rooted trees with 6 vertices.
21 is the smallest number of distinct squares needed to tile a square .
22 is the number of partitions of 8.
23 is the smallest number of integer -sided boxes that tile a box so that no two boxes share a common length.
24 is the largest number divisible by all numbers less than its square root .
25 is the smallest square that can be written as a sum of 2 positive squares .
26 is the only positive number to be directly between a square and a cube .
27 is the largest number that is the sum of the digits of its cube.
28 is the 2nd perfect number .
29 is the 7th Lucas number .
30 is the largest number with the property that all smaller numbers relatively prime to it are prime .
31 is a Mersenne prime.
32 is the smallest non-trivial 5th power.
33 is the largest number that is not a sum of distinct triangular numbers.
34 is the smallest number with the property that it and its neighbors have the same number of divisors
35 is the number of hexominoes .
36 is the smallest non-trivial number which is both square and triangular.
37 is the maximum number of 5 th powers needed to sum to any number.
38 is the last Roman numeral when written lexicographically.
39 is the smallest number which has 3 different partitions into 3 parts with the same product.
40 is the only number whose letters are in alphabetical order.
41 is a value of n so that x 2 + x + n takes on prime values for x = 0, 1, 2, ... n-2.
42 is the 5th Catalan number .
43 is the number of sided 7-iamonds .
44 is the number of derangements of 5 items.
45 is a Kaprekar number .
46 is the number of different arrangements (up to rotation and reflection) of 9 non-attacking queens on a 9×9 chessboard.
47 is the largest number of cubes that cannot tile a cube .
48 is the smallest number with 10 divisors .
49 is the smallest number with the property that it and its neighbors are squareful .
50 is the smallest number that can be written as the sum of of 2 squares in 2 ways.
51 is the 6th Motzkin number .
52 is the 5th Bell number.
53 is the only two digit number that is reversed in hexadecimal .
54 is the smallest number that can be written as the sum of 3 squares in 3 ways.
55 is the largest triangular number in the Fibonacci sequence .
56 is the number of reduced 5×5 Latin squares .
57 = 111 in base 7.
58 is the number of commutative semigroups of order 4.
59 is the number of stellations of an icosahedron .
60 is the smallest number divisible by 1 through 6.
61 is the 3rd secant number .
62 is the smallest number that can be written as the sum of of 3 distinct squares in 2 ways.
63 is the number of partially ordered sets of 5 elements.
64 is the smallest number with 7 divisors .
65 is the smallest number that becomes square if its reverse is either added to or subtracted from it.
66 is the number of 8-iamonds .
67 is the smallest number which is palindromic in bases 5 and 6.
68 is the 2-digit string that appears latest in the decimal expansion of π .
69 is a value of n where n2 and n3 together contain each digit once.
70 is the smallest weird number .
71 divides the sum of the primes less than it.
72 is the maximum number of spheres that can touch another sphere in a lattice packing in 6 dimensions.
73 is the smallest multi-digit number which is one less than twice its reverse.
74 is the number of different non- Hamiltonian polyhedra with a minimum number of vertices.
75 is the number of orderings of 4 objects with ties allowed.
76 is an automorphic number .
77 is the largest number that cannot be written as a sum of distinct numbers whose reciprocals sum to 1.
78 is the smallest number that can be written as the sum of of 4 distinct squares in 3 ways.
79 is a permutable prime.
80 is the smallest number n where n and n+1 are both products of 4 or more primes .
81 is the square of the sum of its digits.
82 is the number of 6-hexes .
83 is the number of strongly connected digraphs with 4 vertices.
84 is the largest order of a permutation of 14 elements.
85 is the largest n for which 1 2+22+3 2+ ... +n 2 = 1+2+3+ ... +m has a solution 86 = 222 in base 6.
87 is the sum of the squares of the first 4 primes .
88 is one of only 2 numbers known whose square has no isolated digits 89 = 81 + 92
90 is the number of degrees in a right angle.
91 is the smallest pseudoprime in base 3.
92 is the number of different arrangements of 8 non-attacking queens on an 8×8 chessboard.
93 = 333 in base 5.
94 is a Smith number .
95 is the number of planar partitions of 10.
96 is the smallest number that can be written as the difference of 2 squares in 4 ways.
97 is the smallest number with the property that its first 3 multiples contain the digit 9.
98 is the smallest number with the property that its first 5 multiples contain the digit 9.
99 is a Kaprekar number .
100 is the smallest square which is also the sum of 4 consecutive cubes [04/03, 08:54] ‪+91 99429 98634‬
101 is the number of partitions of 13.
102 is the smallest number with three different digits.
103 has the property that placing the last digit first gives 1 more than triple it.
104 is the smallest known number of unit line segments that can exist in the plane, 4 touching at every vertex.
105 is the largest number n known with the property that n - 2k is prime for k>1.
106 is the number of trees with 10 vertices.
107 is the exponent of a Mersenne prime.
108 is 3 hyperfactorial.
109 has a 5th root that starts 2.555555....
110 is the smallest number that is the product of two different substrings.
111 is the smallest possible magic constant of a 3×3 magic square of distinct primes.
112 is the side of the smallest square that can be tiled with distinct integer-sided squares.
113 is a permutable prime.
114 = 222 in base 7.
115 is the number of rooted trees with 8 vertices.
116 is a value of n for which n! + 1 is prime.
117 is the smallest possible value of the longest edge in a Heronian Tetrahedron.
118 is the smallest number that has 4 different partitions into 3 parts with the same product.
119 is the smallest number n where either n or n+1 is divisible by the numbers from 1 to 8.
120 is the smallest number to appear 6 times in Pascal's triangle.
121 is the only square of the form 1 + n + n2 + n3 + n4.
122 is the smallest number n>1 so that n concatenated with n-1 0's concatenated with the reverse of n is prime.
123 is the 10th Lucas number.
124 is the smallest number with the property that its first 3 multiples contain the digit 2.
125 is the only number known that contains all its proper divisors as proper substrings 126 = 9C4.
127 is a Mersenne prime.
128 is the largest number which is not the sum of distinct squares.
129 is the smallest number that can be written as the sum of 3 squares in 4 ways.
130 is the number of functions from 6 unlabeled points to themselves.
131 is a permutable prime.
132 is the smallest number which is the sum of all of the 2-digit numbers that can be formed with its digits.
133 is the smallest number n for which the sum of the proper divisors of n divides φ(n).
134 = 8C1 + 8C3 + 8C4.
135 = 11 + 32 + 53.
136 is the sum of the cubes of the digits of the sum of the cubes of its digits.
137 is the smallest prime with 3 distinct digits that remains prime if one of its digits is removed.
138 is a value of n for which n!!! - 1 is prime.
139 is the number of unlabeled topologies with 5 elements.
140 is a harmonic divisor number.
141 is the 6th central trinomial coefficient.
142 is the number of planar graphs with 6 vertices.
143 is the smallest quasi-Carmichael number in base 8.
144 is the largest square in the Fibonacci sequence.
145 is a factorion.
146 = 222 in base 8.
147 is the number of sided 6-hexes.
148 is the number of perfect graphs with 6 vertices.
149 is the smallest number whose square begins with three 2's.
150 = 100101102 = 21124 = 11005, each using 2 different digits an equal number of times.
151 is a palindromic prime.
152 has a square composed of the digits 0-4.
153 is a narcissistic number.
154 is the smallest number which is palindromic in bases 6, 8, and 9.
155 is the sum of the primes between its smallest and largest prime factor.
156 is the number of graphs with 6 vertices.
157 is the smallest number with φ(2n+1) < φ(2n).
158 is the number of planar partitions of 11.
159 is the number of isomers of C11H24.
160 is the number of 9-iamonds.
161 is a Cullen number.
162 is the smallest number that can be written as the sum of of 4 positive squares in 9 ways.
163 is the largest Heegner Number.
164 is the smallest number which is the concatenation of squares in two different ways.
165 is the midpoint of the nth larger prime and nth smaller prime, for 1 ≤ n ≤ 6.
166 is the number of monotone Boolean functions of 4 variables.
167 is the smallest number whose 4th power begins with 4 identical digits
168 is the size of the smallest non-cyclic simple group which is not an alternating group.
162 is the smallest number that can be written as the sum of of 4 positive squares in 9 ways.
163 is the largest Heegner Number.
164 is the smallest number which is the concatenation of squares in two different ways.
165 is the midpoint of the nth larger prime and nth smaller prime, for 1 ≤ n ≤ 6.
166 is the number of monotone Boolean functions of 4 variables.
167 is the smallest number whose 4th power begins with 4 identical digits
168 is the size of the smallest non-cyclic simple group which is not an alternating group.
169 is the 7th Pell number.
170 is the smallest number n for which φ(n) and σ(n) are both square.
171 has the same number of digits in Roman numerals as its cube.
172 = 444 in base 6.
173 has a square containing only 2 digits.
174 is the smallest number that can be written as the sum of of 4 positive distinct squares in 6 ways.
175 = 11 + 72 + 53.
176 is an octagonal pentagonal number.
177 is the number of graphs with 7 edges.
178 has a cube with the same digits as another cube.
179 has a square comprised of the digits 0-4.
180 is the total number of degrees in a triangle.
181 is a strobogrammatic prime.
182 is the number of connected bipartite graphs with 8 vertices.
183 is the smallest number n so that n concatenated with n+1 is square.
184 is a Kaprekar constant in base 3.
185 is the number of conjugacy classes in the automorphism group of the 8 dimensional hypercube.
186 is the number of degree 11 irreducible polynomials over GF(2).
187 is the smallest quasi-Carmichael number in base 7.
188 is the number of semigroups of order 4.
189 is a Kaprekar constant in base 2.
190 is the largest number with the property that it and its ditinct prime factors are palindromic in Roman numerals.
191 is a number n for which n, n+2, n+6, and n+8 are all prime.
192 is the smallest number with 14 divisors.
193 is the largest number that can be written as ab + ac + bc with 0 < a < b < c in a unique way.
194 is the smallest number that can be written as the sum of 3 squares in 5 ways.
195 is the smallest value of n such that 2nCn is divisible by n2.
196 is the smallest number that is not known to reach a palindrome when repeatedly added to its reverse.
197 is a Keith number.
198 = 11 + 99 + 88.
199 is the 11th Lucas number.
200 is the smallest number which can not be made prime by changing one of its digits.