Note: We only allow properties which are not from any human construct (e.g. 10 is the base of our number system; 24 is the number of hours of a day; 666 is the beast number; are not allowed), nor any quality of earth or universe or human (e.g. 3 is the number of spatial dimensions we live in; 7 is the number of continents in the Earth; 46 is the number of human chromosomes; are not allowed), only mathematics-related properties are allowed, and only base-independent properties are allowed (e.g. 13 is the smallest non-palindromic prime; 22 is the smallest multi-digit Smith number; 196 is conjectured to be the smallest Lychael number; are not allowed) (we do not allow properties which are only true in a specific base, but allow properties for the smallest (or largest) base satisfying a condition, or the smallest (or largest) number n satisfying a condition in all base smaller than n). Usually, we choose number-theory related properties. (Specially, we first choose the properties which the number n breaks a law of small numbers, or the number n is the largest number (in the sea of infinity) to satisfy a condition, examples of the former are 23, 43, 94, 102, 561, and examples of the latter are 24, 41, 77, 127 (127 is in fact only conjectured, but very likely to be true)
1 is the multiplicative identity.
2 is the only degree such that no magic squares exist.
4 is the smallest number of colors sufficient to color all planar maps.
5 is the smallest k such that general algebraic equation with degree k cannot be solved algebraically.
6 is the smallest order of non-abelian group. (see OEIS sequence A060652)
7 is the smallest k such that regular k-gon is not constructible using a compass and an unmarked straightedge. (see OEIS sequence A004169)
8 is the smallest positive integer with no primitive roots. (see OEIS sequence A033949)
9 is the only base such that all repunits are triangular numbers (of the form 1+2+3+4+...+n for some n).
10 is the smallest noncototient number. (see OEIS sequence A005278)
11 is the smallest n such that 2^n−1 has more than one primitive prime factor. (see OEIS sequence A086251 and OEIS sequence A161508 and OEIS sequence A143584)
12 appears in the value of the Riemann zeta function at −1, i.e. ζ(−1) = −1/12, which equals the negative value of the reciprocal of 12 (i.e. the additive inverse of the multiplicative inverse of 12), this number is the value counter-intuitively ascribed to the series 1+2+3+4+...
13 is the largest possible prime factor of a number k such that for any positive integers x,y coprime to k, x^x == y (mod k) iff y^y == x (mod k). (see OEIS sequence A320580)
14 is the smallest nontotient number. (see OEIS sequence A005277)
15 is the smallest k>1 such that the number of terms of the k-th cyclotomic polynomial does not equal to the largest prime factor of k. (see OEIS sequence A070537)
16 is the only number of the form a^b = b^a, with a, b nonnegative integers, a ≠ b. (see OEIS sequence A271936)
17 is the largest number n such that there exist n real numbers 0 < a_1, a_2, a_3, ..., a_n < 1 such that all a_i*k are not integers for 1 ≤ i ≤ k and {floor(a_1*k), floor(a_2*k), floor(a_3*k), ..., floor(a_k*k)} = {0, 1, 2, ..., k−1} for all 1 ≤ k ≤ n.
18 is the largest number n such that the set of the integers 0 ≤ x ≤ n−1 and gcd(x,n)=1, does not contain a complete system of residues modulo any prime (such numbers n are {1, 2, 4, 6, 10, 12, 18}). (this set is not in OEIS, but see OEIS sequences A309871, A123985, A007811)
19 is the smallest number n such that Dirichlet character mod n containing numbers whose real and imaginary part are not constructible numbers.
20 is the number of moves (quarter or half turns) required to optimally solve a (3x3x3) Rubik's Cube in the worst case. (see OEIS sequence A257401)
21 is the smallest number of distinct squares needed to tile a square.
22 is the next number n after 15 (and the smallest n if we only count |k| > 1) such that the largest factor of algebraic factor of x^n+-k*y^n has negative coefficient.
23 is the smallest number n such that the relative class number h− of cyclotomic field Q(e^(2πi/(2*n))) is greater than 1. (see OEIS sequence A061653)
24 is the largest number such that 1 is the only quadratic residue coprime to it. (interestingly, if and only if a number n has this property, then n is divisor of 24, see OEIS sequence A018253, this sequence also has many interesting properties of the set of the divisors of 24)
25 is the only base such that all repunits are generalized pentagonal numbers (which appear in the pentagonal number theorem for partition function and sigma function).
26 is the only positive number to be directly between a square and a cube. (related to Pillai conjecture)
27 is a coefficient of the condition of elliptic curves primality proving (i.e. 4*a^3+27*b^2 is not zero, see http://primes.utm.edu/prove/prove4_2.html).
28 is the smallest practical number (OEIS sequence A005153) which is not totient-multiplicative close number (OEIS sequence A301587).
29 is the largest number n such that 2*x^2 + n is prime for all 0≤x≤n−1. (since it is divisible by n for x = n, one cannot do be better than this) (see OEIS sequence A352800 and OEIS sequence A007641)
30 is the largest number with the property that all smaller numbers coprime to it are prime or 1 (such numbers n are {1, 2, 3, 4, 6, 8, 12, 18, 24, 30}). (see OEIS sequence A048597)
31 is the smallest number n such that ʃ(0,∞) (cos(x)cos(x/2)cos(x/3)...cos(x/n)) ≠ π/2.
32 is the smallest number n such that the n-th row of the modulo-2 Pascal's triangle (the top row, which contains only one 1, is the 0th row, not the 1st row), when read in binary, is not a number of the sides of a constructible regular polygon. (see OEIS sequence A001317 and OEIS sequence A045544)
33 is the largest number that is not a sum of distinct triangular numbers.
34 is the smallest nonsquare number n not divisible by 4 with no prime factors p = 3 mod 4 but the period of continued fraction of √n is even. (see OEIS sequence A031398)
35 is the smallest number n>1 such that gcd(n, b^n − b) = 1 for some b. (see OEIS sequence A121707 and OEIS sequence A321487 and OEIS sequence A267999 and OEIS sequence A306097)
36 is the smallest perfect power (OEIS sequence A001597) which is not prime power (OEIS sequence A000961).
37 is the smallest irregular prime. (see OEIS sequence A000928)
38 is the magic constant of the only non-trivial normal magic hexagon.
39 is the smallest number n>2 such that the Mertens function M(n) returns zero. (see OEIS sequence A028442)
40 is the largest number n such that almost all positive integers have 3x+1 sequence reach n.
41 is the largest number n such that x^2 + x + n is prime for all 0≤x≤n−2. (since it is divisible by n for x = n−1, one cannot do be better than this) (see OEIS sequence A014556 and OEIS sequence A056561 and OEIS sequence A005846)
42 is the largest number of sides of a regular polygon which can completely fill a plane vertex with other regular polygons (related to Euclidean tilings by convex regular polygons). (see OEIS sequence A236681 and tetracontadigon)
43 is the smallest number n such that (define a_n: a_0 = 1, for k > 0, a_k = (1+a_0^2+a_1^2+...+a_(k−1)^2)/k) a_n is not integer. (a_n is the Göbel's sequence) (the fractional part of a_43 is 24/43) (see OEIS sequence A003504 and OEIS sequence A259878)
44 is the smallest number n > 2 such that eulerphi(x) = n has exactly 3 solutions. (see OEIS sequence A007367 and OEIS sequence A014197)
45 is the largest number n such that the relative class number h− of cyclotomic field Q(e^(2πi/(2*n))) is 1. (see OEIS sequence A061653 and OEIS sequence A005848)
46 is the largest even number which is a value of n for incrementally largest values of minimal x satisfying the Pell equation x^2−n*y^2=1. (see OEIS sequence A033316)
47 is the largest number of cubes that cannot tile a cube. (see OEIS sequence A353935)
48 is the largest number other than the Mersenne primes such that for all divisors d of n, d+1 is either a prime or a perfect power. (see OEIS sequence A215068)
49 is the smallest number with the property that it and its neighbors are not squarefree. (see OEIS sequence A235578)
50 is the total number of faces of the Platonic solids.
51 is the smallest n such that both n−1 and n+1 are noncototients. (see OEIS sequence A333101)
52 is the smallest untouchable number > 5 (5 is conjectured to be the only odd untouchable number, and if the strong version of the Goldbach conjecture (i.e. every even number > 6 can be written as sum of two distinct primes) is true, then 5 is the only odd untouchable number). (see OEIS sequence A005114)
53 is the smallest prime number that does not divide the order of any sporadic group. (see OEIS sequence A329191)
54 is the smallest totient number which is not totient of squarefree number. (see OEIS sequence A049225 and OEIS sequence A002202)
55 is the largest Fibonacci number which is also a triangular number. (see OEIS sequence A039595)
56 is the only number n such that no x^2 mod n is prime and n is not idoneal number. (see OEIS sequence A065428 and OEIS sequence A000926)
57 is the dimension of the smallest possible homogeneous space for E_8.
58 is the largest squarefree even number n such that the imaginary quadratic field Q(√−n) has class number 2. (see OEIS sequence A005847)
59 is the smallest prime factor of the smallest composite Euclid number (OEIS sequence A006862).
60 is the smallest possible order of nonsolvable group. (see OEIS sequnece A056866 and OEIS sequence A257146 and OEIS sequence A001034)
61 is conjectured to be the largest number n such that k*n−1 and k*n+1 are not both primes for all k ≤ 4*n. (see OEIS sequence A071558)
62 is the smallest n such that the Sum and Product Puzzle has the same solution for range [2,k] for all k ≥ n. (see Sum and Product Puzzle)
63 is the only a^n−b^n (with fixed a and b, variable n) number with n > 2 which have no primitive prime factor (see Zsigmondy's theorem).
64 is the smallest number >1 which can be written as k^p with p prime in 2 different ways. (see OEIS sequence A164345, but 6^6 = 46656 is also such number)
65 is the number of idoneal numbers. (see OEIS sequence A000926)
66 is the denominator of the first Bernoulli number whose absolute value is not a unit fraction (B_10 = 5/66).
67 is the smallest prime which is both Bernoulli irregular and Euler irregular. (see OEIS sequence A128197)
68 is conjecture to be the only n such that there is a prime p > n such that n^(p−1) == 1 (mod p^3). (see OEIS sequence A249275) (reference)
69 is the number in the square root of the algebraic form of the plastic number.
70 is the smallest weird number. (see OEIS sequence A006037)
71 is the degree of Conway's polynomial of the look-and-say sequence. (see OEIS sequence A137275)
72 is the smallest Achilles number. (see OEIS sequence A052486)
73 is the largest squarefree number n such that the quadratic ring O_(Q(√n)) is a Euclidean domain. (see OEIS sequence A048981)
74 is the number of different non-Hamiltonian polyhedra with a minimum number of vertices.
75 is the smallest n such that both n−1 and n+1 are nontotients. (see OEIS sequence A333100)
76 is conjectured to be the smallest n such that there are no prime n-Fibonacci numbers (the n-Fibonacci numbers for n = 1, 2, 3, 4, 5, 6 are A000045, A000129, A006190, A001076, A052918, A005668). (the only smaller n with unknown status is 49).
77 is the largest number that cannot be written as a sum of distinct numbers whose reciprocals sum to 1.
78 is the dimension of the exceptional Lie group E_6.
79 is the smallest prime number p for which the real quadratic field Q[√p] has class number greater than 1.
80 is the number of single-tile moves required to optimally solve a 15 puzzle in the worst case.
81 is the only heptagonal number (beside 0 and 1) which is also a perfect power m^r with r > 2.
82 is the number of 6-hexes. (6-hexes is the smallest possible n-hexes which can contain hole)
83 is the sum of the numbers that are not the sum of distinct triangular numbers.
84 is the smallest number n such that n is neither squarefree nor of the form p^a*q^b with p, q primes, but no simple group with order n exists.
85 is conjectured to be the only number == 1 mod 4 which is neither square not sum of a square (including 0) and a prime. (see OEIS sequence A020495)
86
87 is the smallest imaginary quadratic field with class number 6.
88 is the largest number not regular to 840 (the largest number such that all coprime quadratic residues are squares) such that all coprime quadratic residues are squares.
89 is the smallest prime to start a Cunningham chain of the first kind of ≥6 terms. (note that 2 starts a Cunningham chain of the first kind of 5 terms) (also note that there are no primes ≤ 1000000 to start a Cunningham chain of the first kind of ≥7 terms, the smallest such prime is 1122659)
90 is the smallest number which is value of the Carmichael lambda function but not of the Euler totient function. (see OEIS sequence A270266)
91 is the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed. (see cabtaxi number)
92 is the number of elements in look-and-say sequence containing the digits 1, 2, and 3 only.
93 is the number in the square root of the algebraic form of the supergolden ratio.
94 is the smallest number n>1 such that M(n) is positive, where M is the Mertens function. (see OEIS sequence A002321)
95 is the third-smallest number whose aliquot sequence terminates at 6 (within the sequence {95, 25, 6}).
96 is the only number n beside 2 with the property that not n but n/2 is a value of k for incrementally largest values of groups of order k sets a record. (see OEIS sequence A046059 and OEIS sequence A000001)
97
98 is the only known number n of the form 2*p^2 with odd prime p such that Φ_n(2) is (probable) prime.
99
100 is the smallest Lagado number with more than one factorization into Lagado primes. (a Lagado prime is a Lagado number that is not divisible by any Lagado number between 1 and itself) (the Lagado numbers are the numbers = 1 mod 3)
101 is the number of negated discriminants of orders of imaginary quadratic fields with 1 class per genus. (see OEIS sequence A003171)
102 is the smallest integer n for which (let n = 2^(a_0) × 3^(a_1) × 5^(a_2) × 7^(a_3) × 11^(a_4) × ...) a_0 + a_1*x + a_2*x^2 + a_3*x^3 + a_4*x^4 + ... = 0 has solution, but does not have algebraic solution (i.e. there is no solution in radicals). (see Galois theory)
103 is the smallest Lucas-Wieferich prime associated with the pair (P, Q) = (4, 1). (see OEIS sequence A238490) (the only other known such prime is 2297860813)
104 is the finally period of Langton's ant.
105 is the smallest integer such that the factorization of x^n−1 over Q includes coefficients other than ±1, in other words, the 105th cyclotomic polynomial, Φ_105, is the first with coefficients other than ±1. (see OEIS sequence A013590)
108 is the number of heptominoes (7-minoes). (7-minoes is the smallest possible n-minoes which can contain hole)
109 is the number of different families of subsets of a three-element set whose union includes all three elements.
110 is the smallest number (>2) which is == 2 mod 12 and is a totient. (see OEIS sequence A063668)
111 is the magic constant of the smallest 3×3 magic square using only 1 and prime numbers.
112 is the side of the smallest square that can be tiled with distinct integer-sided squares.
113 is conjectured to be the only prime such that there is b<p such that b^(p−1) == 1 (mod p^3). (see OEIS sequence A249275) (reference)
114 is the smallest multiple of 6 which is not in the range of the totient function.
115 is the smallest n such that the sum of the 5th powers of the proper divisors of n is prime.
116 is the final population for the Conway's game of Life starting with the "F-pentomino".
117 is the smallest possible length of the longest side of a Heronian tetrahedron (one whose sides are all rational numbers).
118 is the smallest n such that the range n, n + 1, ... 4n/3 contains at least one prime from each of these forms: 4k + 1, 4k - 1, 6k + 1 and 6k - 1
119 is the largest number n such that the nth triangular number is also a tetrahedral number. (see OEIS sequence A027568)
120 is the smallest number (>1) which appears ≥5 times in Pascal's triangle (see OEIS sequence A062527 and OEIS sequence A180058 and OEIS sequence A003016 and OEIS sequence A059233)
121 is the only Brazilian number with exactly 3 divisors.
123 is the smallest k for which there is no known prime of the form (k−1)×k^n+1.
124 is the smallest nontotient which is also an untouchable number.
126 is the number of different semigroups on 4 elements (up to isomorphism and reversal).
127 is conjectured to be the largest number such that all three conditions of The New Mersenne Conjecture (n is of the form 2^k±1 or 4^k±3 (or both), 2^n−1 is prime, (2^n+1)/3 is prime) are true. (The New Mersenne Conjecture is that there is no number such that exactly two of these three conditions are true) (see OEIS sequence A107360)
128 is the largest number that is not a sum of distinct square numbers.
130 is the only integer that is the sum of the squares of its first four divisors.
131 is the smallest Sophie Germain prime congruent to 3 mod 4 which is not safe prime.
132 is the number of primary pretenders. (see OEIS sequence A108574)
134 is the smallest number whose aliquot sum is a weird number.
136 is the only number which is a self-descriptive number in some base (base 4) which has a smaller self-descriptive number.
137 is
138 is the smallest n>5 such that both n−1 and n+1 are strictly non-palindromic numbers.
139 is the largest prime factor among the smallest pair of odd amicable numbers.
140 is the largest number whose square is a tetrahedral number. (see OEIS sequence A003556)
141 is the smallest n>1 such that n×2^n+1 is prime (Cullen prime).
142 is the number of planar graphs with 6 unlabeled vertices.
144 is the largest Fibonacci number which is also a square number.
146 is the smallest untouchable number which is semiprime. (also the smallest untouchable number >2 which is = 2 mod 4)
149 is the smallest number which is not sum of two prime powers (including 1).
150 is the smallest unitary admirable number which is not squarefree.
153 is the sum of the first 5 positive factorials.
155 is the sum of the primes between its smallest and largest prime factor.
157 is the smallest irregular prime with irregular index greater than 1. (see OEIS sequence A060974 and OEIS sequence A091888)
158 is the largest base b for which there are no primary pretenders <b. (see OEIS sequence A000790)
163 is the largest Heegner number. (see OEIS sequence A003173)
165 is the midpoint of the nth larger prime and nth smaller prime for all 1≤n≤6.
168 is the smallest possible order of noncyclic simple group which is not alternating groups A_k (which is always a noncyclic simple group for k≥5).
170 is the smallest even number n such that the smallest N such that the multiplicative group of integers modulo N ([Z/NZ]×) as a product of cyclic groups C_k1 × C_k2 × ... × C_km contains a copy of C_n has m > 2. (the corresponding N is 1542013, and the multiplicative group of integers modulo 1542013 ([Z/1542013Z]×) is isomorphic to C_2 × C_170 × C_4080) (see OEIS sequence A302099)
177 is the magic constant of the smallest 3×3 magic square using only prime numbers.
179 is the largest prime factor of the smallest Catalan pseudoprime (and the only known Catalan pseudoprime which is not square of Wieferich prime).
180 is the largest possible number of edges of an Archimedean solid.
181 is the largest value x satisfying the Ramanujan–Nagell equation (2^n−7 = x^2).
188 is the number of semigroups with order 4.
191 is the smallest prime congruent to 1 mod 19 (this is conjectured to be the case which (let a(n) is the smallest k such that kn+1 is prime) log_n(a(n)−1) is largest, it is also conjectured that log_n(a(n)−1) is always < 0.75, note that log_n(a(n)−1) < 0.74 does not work, and log_19(9) is exactly the counterexample, it also has been conjectured that log_n(a(n)−1) < 1). (note that 19 is the smallest primitive root mod 191, this is the second-largest case which smallest primitive root mod p (with p prime) is larger than √p, the largest case is the smallest primitive root mod 409 is 21)
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 number of irreducible representations of the Monster group.
195 is the smallest n such that binomial(2*n, n) is divisible by n^2.
197 is the smallest prime p such that none of 2*p+1, 4*p+1, 8*p+1, 10*p+1, 14*p+1, 16*p+1 are primes (if at least one of them is prime, then Fermat Last Theorem are easily to prove for the exponant p).
199 is the largest k such that all positive values of k−2n^2 are primes or 1.
200 is the largest number n ≤ 100000000 such that |M(n)| ≥ (√n)/2, where M is the Mertens function. (Mertens conjectured that |M(n)| < √n for all n > 1, this is now known to be false)
201 is the smallest number whose square is de Polignac number.
205 is the smallest nonsquare number n not divisible by 4 and with no prime factors p = 3 mod 4 but the period of continued fractions of √n is even.
208 is the smallest number which is neither untouchable nor infinity-touchable (not n-untouchable for any n, the sequence for n-untouchable numbers for n = 1, 2, 3, 4, 5 are A005114, A283152, A284147, A284156, A284187).
209 is the smallest Chebyshev pseudoprime base 2.
210 is the largest number n such that all primes between n/2 and n yield a representation as a sum of two primes.
216 is conjectured to be the only number not of the form t+p, with t triangular number (including 0 and 1), p either prime or 0.
217 is conjectured to be the largest number n such that σ(n)−n is odd but there are no k ≠ n such that σ(k)−k = σ(n)−n.
219 is the number of space groups, not including handedness.
220 is the smallest number which is a member of amicable number pair. (see OEIS sequence A063990)
227 is a quadratic nonresidue mod every number 3≤n≤28, besides, it is also a primitive root mod every number 1≤n≤18 which have a primitive root.
230 is the number of space groups, including handedness.
231 is the smallest number with ≥3 odd prime factors whose cyclotomic polynomial has all coefficients ±1.
237 is the smallest n divisible by 3 such that n*2^k−1 and n*2^k+1 cannot be twin primes.
239 is the only number beside 23 which is not the sum of 8 or fewer cubes.
240 is the kissing number in 8 dimensions. (this is the E_8 lattice) (note that the true value of the kissing number is only known in 1, 2, 3, 4, 8, and 24 dimensions)
246 is the smallest number n for which it is known that there is an infinite number of prime gaps no larger than n (Yi-tang Zhang's result).
247 is the smallest n such that both n−1 and n+1 are untouchable.
255 is the smallest cyclic number which is neither prime nor semiprime. (see OEIS sequence A050384)
256 is conjectured to be the largest number n containing the digit b−1 in no base b with 2 < b < n. (see OEIS sequence A337536)
257 is the only known Fermat prime which is irregular prime.
261 is the smallest number that appear 3 times in Recamán's sequence. (the smallest number that appear 2 times in Recamán's sequence is 42) (see OEIS sequence A005132 and OEIS sequence A064369 and OEIS sequence A057167)
264 is the largest number not regular to "the largest number such that all coprime quadratic residues are prime powers (including 1)" (i.e. 1680) such that all coprime quadratic residues are prime powers (including 1)
271 is the smallest prime p such that neither p−1 nor p+1 is cubefree.
272 is conjectured to be the largest negative number appearing in a cycle of Collatz (3x+1) conjecture.
275 is the smallest nonsemiprime n>1 such that gcd(n, b^n − b) = 1 for some b. (see OEIS sequence A121707 and OEIS sequence A321487 and OEIS sequence A267999 and OEIS sequence A306097)
276 is the smallest number whose aliquot sequence has not yet been fully determined. (see OEIS sequence A131884)
277 is the 8th Euler (or up/down) number.
281 is the largest prime p such that (1!+2!+3!+4!+ ... +p!) − 2 is prime.
283 is the smallest n such that φ^7(n) > 1.
284 is the smallest n appearing twice in P union Q union R defined with: Construct sequences P, Q, R by the rules: Q = first differences of P, R = second differences of P, P starts with 1, 3, 9, Q starts with 2, 6, R starts with 4; at each stage the smallest number not yet present in P, Q, R is appended to R. (see OEIS sequence A225385 and OEIS sequence A225386 and OEIS sequence A225387)
286 is the smallest nonsemiprime which is a possible value of the smallest (Fermat) prime base n.
288 is the smallest n>8 such that both n and n+1 are powerful. (8 is the largest n such that both n and n+1 are perfect powers)
290 is the smallest n such that a positive definite integral quadratic form is universal if it takes the numbers from 1 to n as values.
291 is the largest number that is not the sum of distinct non-trivial powers.
292 is the 5th term of the continued fraction of π.
300 is the smallest nonsquare number which is not a primitive root mod any safe prime.
301 is the smallest 6-hyperperfect number.
305 is the smallest odd number n≥3 such that there are no known prime of the form n^k−2 with k≥1.
306 is the near-integer of π^5.
307 is the smallest prime = 1 mod 3 for which 2 and 3 are cubic residues.
315 is the smallest odd number n such that φ(n) < φ(n−1).
316 is the sum of all Heegner numbers.
318 is conjectured to be the smallest number that start an unbounded aliquot-like sequence based on Dedekind psi function.
323 is the smallest Fibonacci U-pseudoprime. (see OEIS sequence A081264)
324 is the smallest b which is not odd power (see OEIS sequence A070265) but no (b^n+1)/(b+1) prime exists. (see OEIS sequence A084742)
325 is the only known 3-hyperperfect number.
336 is the smallest number n such that the equation n = k*sigma(k) has more than one solution.
337 is the smallest odd number n such that |2^k−n| is composite for all 1≤k≤n.
341 is the smallest (Fermat) pseudoprime base 2 (also called Sarrus number or Poulet number). (see OEIS sequence A001567)
343 is the only cube which is a nontrivial repunit in some base (base 18).
348 is the smallest base such that the smallest (Fermat) pseudoprime greater than the base is Carmichael number (1105).
351 is the smallest n such that n±1 and n±3 are all abundant.
353 is the smallest number whose 4th power can be written as the sum of four 4th powers.
354 is the sum of the absolute values of the coefficients of Conway's polynomial.
360 is the smallest order of simple group which is a multiple of a smaller order of simple group (60).
365 is the smallest number that can be written as a sum of consecutive squares in more than 1 way.
373 is conjectured to be the largest prime p such that (previous prime(p))# ± p are both primes.
379 is the largest known prime p such that p! − 1 and p# + 1 are both primes.
380 is the smallest even base not of the form n^x (where generalized repunits can be factored algebraically) for which no generalized repunit (probable) primes are known.
383 is the largest base b for which there are no primary pretenders < b−1. (see OEIS sequence A000790)
384 is the order of the hyperoctahedral group for n = 4.
385 is the smallest integer such that the factorization of x^n−1 over Q includes coefficients with absolute value greater than 2.
396^2n appears in a denominator of an infinite product of π.
398 is conjectured to be the largest n which is a quadratic nonresidue mod any odd prime < sqrt(n).
399 is the smallest Lucas-Carmichael number. (see OEIS sequence A006972)
400 is conjectured to be the largest perfect power which is a repunit with >2 1's in some base. (see OEIS sequence A208242)
401 is the smallest fundamental discriminant of real quadratic number fields with class number 5.
409 is conjectured to be the largest prime p whose smallest primitive root is larger than √p (see OEIS sequence A262264 and OEIS sequence A001918 and OEIS sequence A002230 and OEIS sequence A023048).
418 is the smallest non-primepower k such that binomial(2*k, k) == 2 (mod k). (besides, 418 is also the only known such even non-primepower k) (see OEIS sequence A328497 and OEIS sequence A082180 and OEIS sequence A228562 and OEIS sequence A136327)
432 is the only number k besides −1 (which corresponds to Catalan conjecture) such that the Mordell curves x^2+k=y^3 (which are specific elliptic curves) has rational solutions with both x and y nonzero, but only finitely many rational solutions (only consider primitive k, i.e. sixth-power-free k).
441 is the smallest Hilbert number with more than one factorization into Hilbert primes. (a Hilbert prime is a Hilbert number that is not divisible by any Hilbert number between 1 and itself) (the Hilbert numbers are the numbers = 1 mod 4)
449 is the number of superabundant numbers which are also highly composite numbers. (see OEIS sequence A166981)
454 is the largest number which is not the sum of 7 or fewer cubes.
455 is the smallest n with omega(n)>2 such that gcd(n, b^n − b) = 1 for some b. (see OEIS sequence A121707 and OEIS sequence A321487 and OEIS sequence A267999 and OEIS sequence A306097)
462 is conjectured to be the largest base b for which there are no (Fermat) pseudoprimes ≤ b+1. (see OEIS sequence A090086)
467 is the smallest prime p which divides (1!+2!+3!+4!+ ... +p!) − 2.
468 is the largest module for the known property of odd perfect numbers. (the known property of odd perfect numbers is = 1 mod 12, or = 117 mod 468, or = 81 mod 324)
473 is the smallest composite k such that 1^(k-1) + 2^(k-1) + 3^(k-1) == 3 (mod k).
480 is the largest number n such that carmichael_lambda(n) = 8.
491 is the smallest irregular prime with irregular index greater than 2.
492 is the smallest number of faces such that holyhedron is known to exist.
500 is the smallest strong Achilles number. (see OEIS sequence A194085)
504 is the largest number n such that carmichael_lambda(n) = 6.
521 is the square root of the smallest Perrin pseudoprime.
533 is the smallest n such that both n−1 and n+1 are both nontotients and noncototients.
536 is the smallest number > e^(2π) (e^(2π) the smallest positive real number x such that x^i = 1)
560 is the smallest number which is a Rhonda number in some base.
561 is the smallest Carmichael number. (see OEIS sequence A002997)
562 is the smallest number not itself an amicable pair whose Aliquot sequence terminates at an amicable pair.
563 is the largest known Wilson prime. (see OEIS sequence A007540)
564 is the smallest number such that it is unknown whether it is in a sociable number cycle. (see OEIS sequence A122726 and OEIS sequence A347770)
572 is conjectured to be the largest base b for which there are no (Fermat) pseudoprimes < b+1. (see OEIS sequence A090086)
580 is the smallest number n such that eulerphi(x) = n has only two solutions and the smaller of this two solutions is not prime power (including 1).
598 is the smallest even quasi-Carmichael number.
600 is the largest possible number of cells of 4-dimension polytope. (see OEIS sequence A063924) (note that for n-dimension polytope, n≥5, the only possible number of cells are n+1, 2*n, and 2^n)
619 is conjectured to be the largest number n containing the digit 0 in no base b with 2 < b < n (see OEIS sequence A069575)
630 is the smallest unitary admirable number which is not primitive unitary abundant number (unitary abundant numbers having no unitary abundant proper unitary divisor).
641 is the smallest prime which is a prime factor of a composite Fermat number.
645 is the smallest nonsemiprime which is not Carmichael number and is a possible value of the Euler primary pretender to base n. (note that the only nonsemiprime which is a possible value of the primary pretender to base n is the smallest Carmichael number (561))
648 is the smallest number which can be written as m*n^m with m,n integers > 1 in two different ways.
657 is conjectured to be the largest number which is not the sum of a semiprime and a square (including 0). (see OEIS sequence A100570)
691 is the first irregular prime to appear in the numerator of a Bernoulli number. (see OEIS sequence A046753 and OEIS sequence A189683)
700 is the smallest base such that the smallest (Fermat) pseudoprime is Carmichael number (561).
703 is the smallest number n which is strong pseudoprime to the smallest base b such that the Jacobi symbol (b|n) is −1.
705 is the smallest Fibonacci V-pseudoprime. (see OEIS sequence A005845)
714 is the largest number n such that n*(n+1) is a primorial.
720 is the smallest number which cannot be written as m*n with both m and n are powers of squarefree numbers (including 1).
744 is the constant term of modular function j as power series in q=e^(2πit). (see OEIS sequence A000521)
771 is the only nonsquare number > 5 not == 1 mod 3 which is not the sum of a square and a prime.
773 is the smallest odd number n such that 2^k+n is composite for all 1≤k≤n.
781 is the period of the sequence of Bell numbers mod 5.
786 is conjectured to be the largest n such that binomial(2*n, n) is not divisible by a square of an odd prime.
807 is conjectured to be the smallest number n divisible by 3 such that n×2^k−1 and n×2^(k+1)−1 are not both primes for all k≥1.
836 is the smallest weird number which is also an untouchable number.
840 is the largest number n such that k^2 mod n is square number for all k coprime to n. (see OEIS sequence A303704)
945 is the smallest odd abundant number. (see OEIS sequence A005231)
946 is conjectured to be the largest even number which is (Fermat) pseudoprime to 1/4 of the bases coprime to it. (see OEIS sequence A247074 and OEIS sequence A063994)
989 is is the smallest extra strong Lucas pseudoprime. (OEIS sequence A217719)
991 is the largest number which cannot be expressed as the sum of abundant numbers.
993 is conjectured to yields the highest residue for 2^h(n)/(3^t(n)*n), where h and t are the number of halving resp. tripling steps in the Collatz problem. (see OEIS sequence A127789)
1012 is conjectured to be the largest base b for which there are no (Fermat) pseudoprimes < b−1. (see OEIS sequence A090086)
1024