2000 (number)

the highest number expressible using only two unmodified characters in Roman numerals (MM)
an Achilles number^[1]
smallest four digit eban number^[2]
the sum of all the nban numbers in the sequence^[3]

Selected numbers in the range 2001–2999

2001 to 2099

2001 – sphenic number^[4]
2002 = 7⁴ – 7³ – 7² – 7. Palindromic number in decimal, base 76, 90, 142, and 11 other non-trivial bases. A binomial coefficient, equal to ${\tbinom {14}{5}}$ .
2003 – Sophie Germain prime and the smallest prime number in the 2000s
2004 – Area of the 24th crystagon^[5]
2005 – A vertically symmetric number
2006 – number of subsets of {1,2,3,4,5,6,7,8,9,10,11} with relatively prime elements^[6]
2007 – 2²⁰⁰⁷ + 2007² is prime^[7]
2008 – number of 4 × 4 matrices with nonnegative integer entries and row and column sums equal to 3^[8]
2009 = 28² + 35², sum of two squares
2010 – number of compositions of 12 into relatively prime parts^[9]
2011 – sexy prime with 2017, sum of eleven consecutive primes: 2011 = 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211
2012 – The number 8 × 10²⁰¹² − 1 is a prime number^[10]
2013 – number of widely totally strongly normal compositions of 17
2014 – 5 × 2²⁰¹⁴ - 1 is prime^[11]
2015 – Lucas–Carmichael number^[12]
2016 – second-smallest Erdős–Nicolas number,^[13] triangular number,^[14] number of 5-cubes in a 9-cube, 2¹¹-2⁵
2017 – Mertens function zero, sexy prime with 2011
2018 – Number of partitions of 60 into prime parts
2019 – smallest number that can be represented as the sum of 3 prime squares 6 different ways: 2019 = 7² + 11² + 43² = 7² + 17² + 41² = 13² + 13² + 41² = 11² + 23² + 37² = 17² + 19² + 37² = 23² + 23² + 31²^[15]
2020 – sum of the totient function for the first 81 integers; Self-descriptive number
2021 = 43 × 47, consecutive prime numbers, next is 2491
2022 – non-isomorphic colorings of a toroidal 3 × 3 grid using exactly three colors under translational symmetry,^[16] beginning of a run of 4 consecutive Niven numbers^[17]
2023 = 7 × 17² – multiple of 7 with digit sum equal to 7,^[18] sum of squares of digits equals 17
2024 – tetrahedral number^[19]
2025 = 45², square of the sum of the first nine positive integers (and therefore sum of the cubes of the first nine positive integers, by Nicomachus's theorem), centered octagonal number,^[20] lowest number with exactly 15 odd divisors.^[21] Sum of odd numbers from 1 to 89.
2026 = Number of hyperforests spanning 10 unlabeled nodes without isolated vertices^[22]
2027 – super-prime, safe prime^[23]
2028 = 13³ – 13²
2029 – member of the Mian–Chowla sequence^[24]
2030 = 21² + 22² + 23² + 24² = 25² + 26² + 27²
2031 – centered pentagonal number^[25]
2032 – number of binary Lyndon words of length 16 with an even number of 1's^[26]
2033 – number of rooted trees with 9 nodes and a single labeled node^[27]
2034 – number of unlabeled graphs on 11 nodes whose components are unicyclic graphs^[28]
2035 – Wolstenholme number^[29]
2036 – Eulerian number^[30]
2037 = 2¹¹ - 11^[31]
2038 – Number of unlabeled Euler graphs with 9 nodes^[32]
2039 – Sophie Germain prime, safe prime^[23]
2040 = ${\frac {15\times 16\times 17}{2}}$ ^[33]
2041 – Number of 11-node connected graphs with at most one cycle^[34]
2042 = 2 × 1021. All the digits of all the prime factors are smaller than 3^[35]
2043 – Number of partitions of 35 in which the number of parts divides 35^[36]
2044 = $\sigma _{3}(12)=\sum _{d|12}d^{3}$ ^[37]
2045 – Number of partially ordered set with 7 unlabeled elements^[38]
2046 = 2¹¹ - 2 = the expected number of tosses of a fair coin to get 10 consecutive heads^[39]
2047 – super-Poulet number,^[40] Woodall number,^[41] decagonal number,^[42] a centered octahedral number,^[43] 2047 = 2¹¹ - 1 = 23 × 89 and is the first Mersenne number that is composite for a prime exponent
2048 = 2¹¹
2049 = 2¹¹ + 2⁰. A sum of two positive powers of two^[44]
2050 = 31² + 33². Sum of 2 consecutive odd squares
2051 = 1⁵ + 1⁵ + 1⁵ + 4⁵ + 4⁵. Sum of 5 positive 5th powers^[45]
2052 = 2¹¹ + 2². A sum of two positive powers of two
2053 – star number
2054 = 1⁹ + 1⁹ + 1⁹ + 1⁹ + 1⁹ + 1⁹ + 2⁹ + 2⁹ + 2⁹ + 2⁹. Sum of 10 positive 9th powers
2055 = 1¹⁰ + 1¹⁰ + 1¹⁰ + 1¹⁰ + 1¹⁰ + 1¹⁰ + 1¹⁰ + 2¹⁰ + 2¹⁰. Sum of 9 positive 10th powers
2056 – magic constant of n × n normal magic square and n-queens problem for n = 16
2057 = 1¹⁰ + 1¹⁰ + 1¹⁰ + 1¹⁰ + 1¹⁰ + 1¹⁰ + 1¹⁰ + 1¹⁰ + 1¹⁰ + 2¹⁰ + 2¹⁰. Sum of 11 positive 10th powers
2058 = $49\times \phi (49)$ ^[46]
2059 = 3⁷-2⁷^[47]
2060 – sum of the totient function for the first 82 integers
2061 – Number of sets of positive integers with arithmetic mean 7^[48]
2062 = $\phi (\phi (2062)+\sigma (2062))$ ^[49]
2063 – Sophie Germain prime, safe prime,^[23] super-prime
2064 = 1031 + 1033, which is a twin prime sum^[50]
2065 = Number of distinct lines through the origin in the fourdimensional lattice of side length 6^[51]
2066 – Bell number^[52]
2067 = Number of Golomb partitions of 30^[53]
2068 – number of 16-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed^[54]
2069 – Sophie Germain prime
2070 – pronic number^[55]
2071 = Number of sensed planar maps with 6 edges^[56]
2072 = 45² + 45 + 2^[57]
2073 – Genocchi number^[58]
2074 = Number of Baxter permutations of length 7^[59]
2075 = 411 + 413 + 415 + 417 + 419 = 25 + 50×41^[60]
2076 = Number of disconnected regular graphs with 17 nodes^[61]
2077 = Number of canonical polygons with 16 edges having 2-fold rotational symmetry^[62]
2078 = Number of reversible strings with 12 beads using exactly two different colors^[63]
2079 = ${\frac {9\cdot 10\cdot 11\cdot 12\cdot (2\cdot 9+3)}{5!}}$ , 5-dimensional pyramidal number^[64]
2080 – triangular number
2081 – super-prime, first member of a prime quadruplet
2082 = 2¹¹+2⁵+2¹^[65]
2083 – second member of a prime quadruplet
2084 = $\lfloor 4\times \phi ^{13}\rfloor$ ,^[66] where $\phi$ is the golden ratio
2085 – average of a prime quadruplet
2087 – third member of a prime quadruplet
2089 – fourth member of a prime quadruplet
2093 – Mertens function zero
2095 – Mertens function zero
2096 – Mertens function zero
2097 – Mertens function zero
2099 – Mertens function zero, super-prime, safe prime,^[23] highly cototient number^[67]

2100 to 2199

2100 – Mertens function zero
2101 – centered heptagonal number^[68]
2107 – member of a Ruth–Aaron pair with 2108 (first definition)
2108 – member of a Ruth–Aaron pair with 2107 (first definition)
2109 – square pyramidal number,^[69] the sum of the third and last trio of three-digit permutable primes in decimal: 199 + 919 + 991
2112 – The break-through album of the band Rush
2113 – Mertens function zero, Proth prime,^[70] centered square number^[71]
2116 = 46²
2117 – Mertens function zero
2119 – Mertens function zero
2120 – Mertens function zero, Fine number^[72]
2122 – Mertens function zero
2125 – nonagonal number^[73]
2127 – sum of the first 34 primes
2129 – Sophie Germain prime
2135 – Mertens function zero
2136 – Mertens function zero
2137 – prime of the form 2p-1
2138 – Mertens function zero
2141 – Sophie Germain prime
2142 – sum of the totient function for the first 83 integers
2143 – almost exactly 22 $π$ ⁴
2145 – triangular number
2153 – with 2161, smallest consecutive primes that have the same sum of digits as each other's prime indices
2160 – largely composite number^[74]
2161 – with 2153, smallest consecutive primes that have the same sum of digits as each other's prime indices
2162 – pronic number^[55]
2166 – sum of the totient function for the first 84 integers
2169 – Leyland number^[75]
2171 – Mertens function zero
2172 – Mertens function zero
2175 – smallest number requiring 143 seventh powers for Waring representation
2176 – pentagonal pyramidal number,^[76] centered pentagonal number,^[25] number of prime knots with 12 crossings
2178 – first natural number whose digits in its decimal representation get reversed when multiplied by 4^[77]
2179 – Wedderburn–Etherington prime^[78]
2184 – equals both 3⁷ − 3 and 13³ − 13 and is believed to be the only such doubly strictly absurd number^[79]^{[unreliable source?]}
2187 = 3⁷, vampire number,^[80] perfect totient number^[81]
2188 – Motzkin number^[82]
2197 = 13³, palindromic in base 12 (1331₁₂)
2199 – perfect totient number^[81]

2200 to 2299

2201 – only known non-palindromic number whose cube is palindromic; also no known fourth or higher powers are palindromic for non-palindromic numbers
2203 – Mersenne prime exponent
2205 – odd abundant number^[83]
2207 – safe prime,^[23] Lucas prime^[84]
2208 – Keith number^[85]
2209 = 47², palindromic in base 14 (B3B₁₄), centered octagonal number^[20]
2211 – triangular number
2221 – super-prime, happy number
2222 – repdigit
2223 – Kaprekar number^[86]
2230 – sum of the totient function for the first 85 integers
2232 – decagonal number^[42]
2236 – Harshad number
2245 – centered square number^[71]
2248 – 2¹¹ + 200
2254 – member of the Mian–Chowla sequence^[24]
2255 – octahedral number^[87]
2256 – pronic number^[55]
2269 – super-prime, cuban prime^[88]
2272 – sum of the totient function for the first 86 integers
2273 – Sophie Germain prime
2276 – sum of the first 35 primes, centered heptagonal number^[68]
2278 – triangular number
2281 – star number, Mersenne prime exponent
2287 – balanced prime^[89]
2294 – Mertens function zero
2295 – Mertens function zero
2296 – Mertens function zero
2299 – member of a Ruth–Aaron pair with 2300 (first definition)

2300 to 2399

2300 – tetrahedral number,^[19] member of a Ruth–Aaron pair with 2299 (first definition)
2301 – nonagonal number^[73]
2304 = 48²
2306 – Mertens function zero
2309 – primorial prime, twin prime with 2311, Mertens function zero, highly cototient number^[67]
2310 – fifth primorial^[90]
2311 – primorial prime, twin prime with 2309
2321 – Mertens function zero
2322 – Mertens function zero
2326 – centered pentagonal number^[25]
2328 – sum of the totient function for the first 87 integers, the number of groups of order 128^[91]
2331 – centered cube number^[92]
2338 – Mertens function zero
2339 – Sophie Germain prime, twin prime with 2341
2341 – super-prime, twin prime with 2339
2346 – triangular number
2347 – sum of seven consecutive primes (313 + 317 + 331 + 337 + 347 + 349 + 353)
2351 – Sophie Germain prime, super-prime
2352 – pronic number^[55]
2357 – Smarandache–Wellin prime^[93]
2368 – sum of the totient function for the first 88 integers
2372 – logarithmic number^[94]
2378 – Pell number^[95]
2379 – member of the Mian–Chowla sequence^[24]
2381 – super-prime, centered square number^[71]
2393 – Sophie Germain prime
2397 – sum of the squares of the first ten primes
2399 – Sophie Germain prime

2400 to 2499

2400 – perfect score on SAT tests administered after 2005
2401 = 49² = 7⁴, centered octagonal number^[20]
2402 – 7⁴ + 1
2406 appears for the first time in the Recamán's sequence at n = 394,178,473,633,984. Or in other words A057167(2406) = 394,178,473,633,984
2415 – triangular number
2417 – super-prime, balanced prime^[89]
2425 – decagonal number^[42]
2427 – sum of the first 36 primes
2431 – product of three consecutive primes
2437 – cuban prime,^[88] largest right-truncatable prime in base 5
2447 – safe prime^[23]
2450 – pronic number^[55]
2456 – sum of the totient function for the first 89 integers
2458 – centered heptagonal number^[68]
2459 – Sophie Germain prime, safe prime^[23]
2465 – magic constant of n × n normal magic square and n-queens problem for n = 17, Carmichael number^[96]
2470 – square pyramidal number^[69]
2471 – number of ways to partition {1,2,3,4,5,6} and then partition each cell (block) into subcells^[97]
2477 – super-prime, cousin prime
2480 – sum of the totient function for the first 90 integers
2481 – centered pentagonal number^[25]
2484 – nonagonal number^[73]
2485 – triangular number, number of planar partitions of 13^[98]
2491 = 47 * 53, consecutive prime numbers, member of Ruth–Aaron pair with 2492 under second definition
2492 – member of Ruth–Aaron pair with 2491 under second definition

2500 to 2599

2500 = 50², palindromic in base 7 (10201₇)
2501 – Mertens function zero
2502 – Mertens function zero
2503 – Friedman prime
2504 – Friedman number
2505 – Friedman number
2506 – Friedman number
2507 – Friedman number
2508 – Friedman number
2509 – Friedman number
2510 – member of the Mian–Chowla sequence^[24]
2513 – member of the Padovan sequence^[99]
2517 – Mertens function zero
2519 – the smallest number congruent to 1 (mod 2), 2 (mod 3), 3 (mod 4), ..., 9 (mod 10)
2520 – superior highly composite number; smallest number divisible by numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 12; colossally abundant number; Harshad number in several bases. It is also the highest number with more divisors than any number less than double itself (sequence A072938 in the OEIS). Not only it is the 7th (and last) number with more divisors than any number double itself but is also the 7th number that is highly composite and the lowest common multiple of a consecutive set of integers from 1 (sequence A095921 in the OEIS) which is a property the previous number with this pattern of divisors does not have (360). That is, although 360 and 2520 both have more divisors than any number twice themselves, 2520 is the lowest number divisible by both 1 to 9 and 1 to 10, whereas 360 is not the lowest number divisible by 1 to 6 (which 60 is) and is not divisible by 1 to 7 (which 420 is). It is also the 6th and largest highly composite number that is a divisor of every higher highly composite number (sequence A106037 in the OEIS).
2521 – star prime, centered square number^[71]
2522 – Mertens function zero
2523 – Mertens function zero
2524 – Mertens function zero
2525 – Mertens function zero
2530 – Mertens function zero, Leyland number^[75]
2533 – Mertens function zero
2537 – Mertens function zero
2538 – Mertens function zero
2543 – Sophie Germain prime, sexy prime with 2549
2548 = 14³ - 14²
2549 – Sophie Germain prime, super-prime, sexy prime with 2543
2550 – pronic number^[55]
2552 – sum of the totient function for the first 91 integers
2556 – triangular number
2567 – Mertens function zero
2568 – Mertens function zero, number of digits in the decimal expansion of 1000!, or the product of all natural numbers from 1 to 1000
2570 – Mertens function zero
2579 – safe prime^[23]
2580 – Keith number,^[85] forms a column on a telephone or PIN pad
2584 – Fibonacci number,^[100] sum of the first 37 primes
2592 – 3-smooth number (2⁵×3⁴)
2596 – sum of the totient function for the first 92 integers

2600 to 2699

2600 – tetrahedral number,^[19] member of a Ruth–Aaron pair with 2601 (first definition)
- 2600 Hz is the tone used by a blue box to defeat toll charges on long distance telephone calls
- 2600: The Hacker Quarterly is a magazine named after the above
- The Atari 2600 was a popular video game console
2601 = 51², member of a Ruth–Aaron pair with 2600 (first definition)
2609 – super-prime
2620 – telephone number, amicable number with 2924
2625 = a centered octahedral number^[43]
2626 – decagonal number^[42]
2628 – triangular number
2632 – number of consecutive baseball games played by Cal Ripken Jr.
2633 – sum of twenty-five consecutive primes (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)
2641 – centered pentagonal number^[25]
2647 – super-prime, centered heptagonal number^[68]
2652 – pronic number^[55]
2656 – sum of the totient function for the first 93 integers
2665 – centered square number^[71]
2674 – nonagonal number^[73]
2677 – balanced prime^[89]
2680 – number of 11-queens problem solutions
2683 – super-prime
2689 – Mertens function zero, Proth prime^[70]
2693 – Sophie Germain prime
2699 – Sophie Germain prime

2700 to 2799

2701 – triangular number, super-Poulet number^[40]
2702 – sum of the totient function for the first 94 integers
2704 = 52²
2707 – strong prime, model number for the concept supersonic airliner Boeing 2707
2719 – super-prime, largest known odd number which cannot be expressed in the form x² + y² + 10z² where x, y and z are integers.^[101] In 1997 it was conjectured that this is also the largest such odd number.^[102] It is now^[when?] known this is true if the generalized Riemann hypothesis is true.^[103]
2728 – Kaprekar number^[86]
2729 – highly cototient number^[67]
2731 – the only Wagstaff prime with four digits,^[104] Jacobsthal prime
2736 – octahedral number^[87]
2741 – Sophie Germain prime, 400th prime number
2744 = 14³, palindromic in base 13 (1331₁₃)
2747 – sum of the first 38 primes
2749 – super-prime, cousin prime with 2753
2753 – Sophie Germain prime, Proth prime^[70]
2756 – pronic number^[55]
2774 – sum of the totient function for the first 95 integers
2775 – triangular number
2780 – member of the Mian–Chowla sequence^[24]
2783 – member of a Ruth–Aaron pair with 2784 (first definition)
2784 – member of a Ruth–Aaron pair with 2783 (first definition)
2791 – cuban prime^[88]

2800 to 2899

2801 – first base 7 repunit prime
2803 – super-prime
2806 – centered pentagonal number,^[25] sum of the totient function for the first 96 integers
2809 = 53², centered octagonal number^[20]
2813 – centered square number^[71]
2816 – number of parts in all compositions of 10^[105]
2819 – Sophie Germain prime, safe prime, sum of seven consecutive primes (383 + 389 + 397 + 401 + 409 + 419 + 421)^[23]
2821 – Carmichael number^[96]
2835 – odd abundant number,^[83] decagonal number^[42]
2843 – centered heptagonal prime^[106]
2850 – triangular number
2862 – pronic number^[55]
2870 – square pyramidal number^[69]
2871 – nonagonal number^[73]
2872 – tetranacci number^[107]
2875 – number of lines on a quintic threefold^[108]
2879 – safe prime^[23]
2897 – super-prime, Markov prime^[109]

2900 to 2999

2902 – sum of the totient function for the first 97 integers
2903 – Sophie Germain prime, safe prime,^[23] balanced prime^[89]
2909 – super-prime
2914 – sum of the first 39 primes
2915 – Lucas–Carmichael number^[12]
2916 = 54²
2924 – amicable number with 2620
2925 – magic constant of n × n normal magic square and n-queens problem for n = 18, tetrahedral number,^[19] member of the Mian-Chowla sequence^[24]
2926 – triangular number
2939 – Sophie Germain prime
2944 – sum of the totient function for the first 98 integers
2963 – Sophie Germain prime, safe prime, balanced prime^[89]
2964 – number of parallelogram polyominoes with 11 cells^[110]
2965 – greater of second pair of Smith brothers, centered square number^[71]
2969 – Sophie Germain prime
2970 – harmonic divisor number,^[111] pronic number^[55]
2976 – centered pentagonal number^[25]
2988 – number of reduced trees with 20 nodes^[112]
2989 – in hexadecimal, reads as "BAD"
2997 – 1000-gonal number^[113]
2999 – safe prime

Prime numbers

There are 127 prime numbers between 2000 and 3000:^[114]^[115]

2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999