2000 (number)
(Redirected from 2081 (number))
| ||||
|---|---|---|---|---|
| Cardinal | two thousand | |||
| Ordinal | 2000th (two thousandth) | |||
| Factorization | 24 × 53 | |||
| Greek numeral | ,Β´ | |||
| Roman numeral | MM | |||
| Unicode symbol(s) | MM, mm | |||
| Binary | 111110100002 | |||
| Ternary | 22020023 | |||
| Senary | 131326 | |||
| Octal | 37208 | |||
| Duodecimal | 11A812 | |||
| Hexadecimal | 7D016 | |||
| Armenian | Ս | |||
| Egyptian hieroglyph | 𓆽 | |||
Look up two thousand in Wiktionary, the free dictionary.
2000 (two thousand) is a natural number following 1999 and preceding 2001.
It is:
- 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
[edit]2001 to 2099
[edit]- 2001 – sphenic number[4]
- 2002 – palindromic number in decimal, base 76, 90, 142, and 11 other non-trivial bases
- 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 X 4 matrices with nonnegative integer entries and row and column sums equal to 3[8]
- 2009 = 74 − 73 − 72
- 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 – triangular number, number of 5-cubes in a 9-cube, Erdős–Nicolas number,[13] 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[14]
- 2020 – sum of the totient function for the first 81 integers
- 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,[15] beginning of a run of 4 consecutive Niven numbers[16]
- 2023 = 7 * 172 – multiple of 7 with digit sum equal to 7,[17] sum of squares of digits equals 17
- 2024 – tetrahedral number,[18] number of the current calendar year
- 2025 = 452, sum of the cubes of the first nine positive integers (and therefore square of the sum of the first nine positive integers), centered octagonal number;[19] least number with 15 odd divisors[20]
- 2026 = Number of hyperforests spanning 10 unlabeled nodes without isolated vertices[21]
- 2027 – super-prime, safe prime[22]
- 2028 = 133 – 132
- 2029 – member of the Mian–Chowla sequence[23]
- 2030 = 212 + 222 + 232 + 242 = 252 + 262 + 272
- 2031 – centered pentagonal number[24]
- 2032 = number of binary Lyndon words of length 16 with an even number of 1's[25]
- 2033 = number of rooted trees with 9 nodes and a single labeled node[26]
- 2034 = number of unlabeled graphs on 11 nodes whose components are unicyclic graphs[27]
- 2035 – Wolstenholme number[28]
- 2036 – Eulerian number[29]
- 2039 – Sophie Germain prime, safe prime[22]
- 2045 – number of partially ordered set with 7 unlabeled elements[30]
- 2047 – super-Poulet number,[31] Woodall number,[32] decagonal number,[33] a centered octahedral number,[34] 2047 = 211 - 1 = 23 × 89 and is the first Mersenne number that is composite for a prime exponent
- 2048 = 211
- 2050 – sum of 2 consecutive odd squares (31² + 33²)
- 2053 – star number
- 2056 – magic constant of n × n normal magic square and n-queens problem for n = 16
- 2060 – sum of the totient function for the first 82 integers
- 2063 – Sophie Germain prime, safe prime,[22] super-prime
- 2068 – number of 16-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed[35]
- 2069 – Sophie Germain prime
- 2070 – pronic number[36]
- 2080 – triangular number
- 2081 – super-prime
- 2093 – Mertens function zero
- 2095 – Mertens function zero
- 2096 – Mertens function zero
- 2097 – Mertens function zero
- 2099 – Mertens function zero, super-prime, safe prime,[22] highly cototient number[37]
2100 to 2199
[edit]- 2100 – Mertens function zero
- 2101 – centered heptagonal number[38]
- 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,[39] 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,[40] centered square number[41]
- 2116 = 462
- 2117 – Mertens function zero
- 2119 – Mertens function zero
- 2120 – Mertens function zero, Fine number[42]
- 2122 – Mertens function zero
- 2125 – nonagonal number[43]
- 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 22π4
- 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[44]
- 2161 – with 2153, smallest consecutive primes that have the same sum of digits as each other's prime indices
- 2162 – pronic number[36]
- 2166 – sum of the totient function for the first 84 integers
- 2169 – Leyland number[45]
- 2171 – Mertens function zero
- 2172 – Mertens function zero
- 2175 – smallest number requiring 143 seventh powers for Waring representation
- 2176 – pentagonal pyramidal number,[46] centered pentagonal number,[24] number of prime knots with 12 crossings
- 2178 – first natural number whose digits in its decimal representation get reversed when multiplied by 4[47]
- 2179 – Wedderburn–Etherington prime[48]
- 2184 – equals both 37 − 3 and 133 − 13 and is believed to be the only such doubly strictly absurd number[49][unreliable source?]
- 2187 = 37, vampire number,[50] perfect totient number[51]
- 2188 – Motzkin number[52]
- 2197 = 133, palindromic in base 12 (133112)
- 2199 – perfect totient number[51]
2200 to 2299
[edit]- 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[53]
- 2207 – safe prime,[22] Lucas prime[54]
- 2208 – Keith number[55]
- 2209 = 472, palindromic in base 14 (B3B14), centered octagonal number[19]
- 2211 – triangular number
- 2221 – super-prime, happy number
- 2222 – repdigit
- 2223 – Kaprekar number[56]
- 2230 – sum of the totient function for the first 85 integers
- 2232 – decagonal number[33]
- 2236 – Harshad number
- 2245 – centered square number[41]
- 2254 – member of the Mian–Chowla sequence[23]
- 2255 – octahedral number[57]
- 2256 – pronic number[36]
- 2269 – super-prime, cuban prime[58]
- 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[38]
- 2278 – triangular number
- 2281 – star number, Mersenne prime exponent
- 2287 – balanced prime[59]
- 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
[edit]- 2300 – tetrahedral number,[18] member of a Ruth–Aaron pair with 2299 (first definition)
- 2301 – nonagonal number[43]
- 2304 = 482
- 2306 – Mertens function zero
- 2309 – primorial prime, twin prime with 2311, Mertens function zero, highly cototient number[37]
- 2310 – fifth primorial[60]
- 2311 – primorial prime, twin prime with 2309
- 2321 – Mertens function zero
- 2322 – Mertens function zero
- 2326 – centered pentagonal number[24]
- 2328 – sum of the totient function for the first 87 integers, the number of groups of order 128[61]
- 2331 – centered cube number[62]
- 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[36]
- 2357 – Smarandache–Wellin prime[63]
- 2368 – sum of the totient function for the first 88 integers
- 2372 – logarithmic number[64]
- 2378 – Pell number[65]
- 2379 – member of the Mian–Chowla sequence[23]
- 2381 – super-prime, centered square number[41]
- 2383 (2384) – number of delegates required to win the 2016 Democratic Party presidential primaries (out of 4051)
- 2393 – Sophie Germain prime
- 2397 – sum of the squares of the first ten primes
- 2399 – Sophie Germain prime
2400 to 2499
[edit]- 2400 – perfect score on SAT tests administered after 2005
- 2401 = 492 = 74, centered octagonal number[19]
- 2415 – triangular number
- 2417 – super-prime, balanced prime[59]
- 2425 – decagonal number[33]
- 2427 – sum of the first 36 primes
- 2431 – product of three consecutive primes
- 2437 – cuban prime,[58] largest right-truncatable prime in base 5
- 2447 – safe prime[22]
- 2450 – pronic number[36]
- 2456 – sum of the totient function for the first 89 integers
- 2458 – centered heptagonal number[38]
- 2459 – Sophie Germain prime, safe prime[22]
- 2465 – magic constant of n × n normal magic square and n-queens problem for n = 17, Carmichael number[66]
- 2470 – square pyramidal number[39]
- 2471 – number of ways to partition {1,2,3,4,5,6} and then partition each cell (block) into subcells[67]
- 2477 – super-prime, cousin prime
- 2480 – sum of the totient function for the first 90 integers
- 2481 – centered pentagonal number[24]
- 2484 – nonagonal number[43]
- 2485 – triangular number, number of planar partitions of 13[68]
- 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
[edit]- 2500 = 502, palindromic in base 7 (102017)
- 2501 – Mertens function zero
- 2502 – Mertens function zero
- 2503 – Friedman prime
- 2510 – member of the Mian–Chowla sequence[23]
- 2513 – member of the Padovan sequence[69]
- 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[41]
- 2522 – Mertens function zero
- 2523 – Mertens function zero
- 2524 – Mertens function zero
- 2525 – Mertens function zero
- 2530 – Mertens function zero, Leyland number[45]
- 2533 – Mertens function zero
- 2537 – Mertens function zero
- 2538 – Mertens function zero
- 2543 – Sophie Germain prime, sexy prime with 2549
- 2549 – Sophie Germain prime, super-prime, sexy prime with 2543
- 2550 – pronic number[36]
- 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[22]
- 2580 – Keith number,[55] forms a column on a telephone or PIN pad
- 2584 – Fibonacci number,[70] 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
[edit]- 2600 – tetrahedral number,[18] 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[34]
- 2626 – decagonal number[33]
- 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[24]
- 2647 – super-prime, centered heptagonal number[38]
- 2652 – pronic number[36]
- 2656 – sum of the totient function for the first 93 integers
- 2665 – centered square number[41]
- 2674 – nonagonal number[43]
- 2677 – balanced prime[59]
- 2680 – number of 11-queens problem solutions
- 2683 – super-prime
- 2689 – Mertens function zero, Proth prime[40]
- 2693 – Sophie Germain prime
- 2699 – Sophie Germain prime
2700 to 2799
[edit]- 2701 – triangular number, super-Poulet number[31]
- 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.[71] In 1997 it was conjectured that this is also the largest such odd number.[72] It is now[when?] known this is true if the generalized Riemann hypothesis is true.[73]
- 2728 – Kaprekar number[56]
- 2729 – highly cototient number[37]
- 2731 – the only Wagstaff prime with four digits,[74] Jacobsthal prime
- 2736 – octahedral number[57]
- 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[40]
- 2756 – pronic number[36]
- 2774 – sum of the totient function for the first 95 integers
- 2775 – triangular number
- 2780 – member of the Mian–Chowla sequence[23]
- 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[58]
2800 to 2899
[edit]- 2801 – first base 7 repunit prime
- 2803 – super-prime
- 2806 – centered pentagonal number,[24] sum of the totient function for the first 96 integers
- 2809 = 532, centered octagonal number[19]
- 2813 – centered square number[41]
- 2816 – number of parts in all compositions of 10[75]
- 2819 – Sophie Germain prime, safe prime, sum of seven consecutive primes (383 + 389 + 397 + 401 + 409 + 419 + 421)[22]
- 2821 – Carmichael number[66]
- 2835 – odd abundant number,[53] decagonal number[33]
- 2843 – centered heptagonal prime[76]
- 2850 – triangular number
- 2862 – pronic number[36]
- 2870 – square pyramidal number[39]
- 2871 – nonagonal number[43]
- 2872 – tetranacci number[77]
- 2875 — number of lines on a quintic threefold[78]
- 2879 – safe prime[22]
- 2897 – super-prime, Markov prime[79]
2900 to 2999
[edit]- 2902 – sum of the totient function for the first 97 integers
- 2903 – Sophie Germain prime, safe prime,[22] balanced prime[59]
- 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,[18] member of the Mian-Chowla sequence[23]
- 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[59]
- 2964 – number of parallelogram polyominoes with 11 cells[80]
- 2965 – greater of second pair of Smith brothers, centered square number[41]
- 2969 – Sophie Germain prime
- 2970 – harmonic divisor number,[81] pronic number[36]
- 2976 – centered pentagonal number[24]
- 2988 – number of reduced trees with 20 nodes[82]
- 2989 – in hexadecimal, reads as "BAD"
- 2997 – 1000-gonal number[83]
- 2999 – safe prime
Prime numbers
[edit]There are 127 prime numbers between 2000 and 3000:[84][85]
- 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
References
[edit]- ^ Sloane, N. J. A. (ed.). "Sequence A052486 (Achilles numbers - powerful but imperfect: if n = Product(p_i^e_i) then all e_i > 1 (i.e., powerful), but the highest common factor of the e_i is 1, i.e., not a perfect power)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A006933 ('Eban' numbers (the letter 'e' is banned!))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A008537 (Numbers that do not contain the letter 'n'))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A007304 (Sphenic numbers: products of 3 distinct primes))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A022264 (n*(7*n - 1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A085945 (Number of subsets of {1,2,...,n} with relatively prime elements)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A064539 (Numbers n such that 2^n + n^2 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A001496 (Number of 4 X 4 matrices with nonnegative integer entries and row and column sums equal to n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A000740 (Number of 2n-bead balanced binary necklaces of fundamental period 2n, equivalent to reversed complement)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A056721 (Numbers n such that 8*10^n-1 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A001770 (Numbers k such that 5*2^k - 1 is prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A006972 (Lucas-Carmichael numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A194472 (Erdős-Nicolas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ "Can you solve it? 2019 in numbers". the Guardian. 2018-12-31. Retrieved 2021-09-19.
- ^ Sloane, N. J. A. (ed.). "Sequence A294685 (non-isomorphic colorings of a toroidal n X k grid using exactly three colors under translational symmetry)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A141769 (Beginning of a run of 4 consecutive Niven (or Harshad) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A063416 (Multiples of 7 whose sum of digits is equal to 7)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b c d Sloane, N. J. A. (ed.). "Sequence A000292 (Tetrahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b c d Sloane, N. J. A. (ed.). "Sequence A016754 (Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A038547 (Least number with exactly n odd divisors.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A144959 (A134955(n) - A134955(n-1). Number of hyperforests spanning n unlabeled nodes without isolated vertices.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b c d e f g h i j k Sloane, N. J. A. (ed.). "Sequence A005385 (Safe primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b c d e f Sloane, N. J. A. (ed.). "Sequence A005282 (Mian-Chowla sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b c d e f g Sloane, N. J. A. (ed.). "Sequence A005891 (Centered pentagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A051841 (Number of binary Lyndon words with an even number of 1's)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A000107 (Number of rooted trees with n nodes and a single labeled node; pointed rooted trees; vertebrates)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A137917 (number of unlabeled graphs on n nodes whose components are unicyclic graphs)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A007408 (Wolstenholme numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A000295 (Eulerian numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A000112 (Number of partially ordered sets (posets) with n unlabeled elements)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A050217 (Super-Poulet numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A003261 (Woodall numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b c d e Sloane, N. J. A. (ed.). "Sequence A001107 (10-gonal (or decagonal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A001845 (Centered octahedral numbers (crystal ball sequence for cubic lattice))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A000013 (Definition (1): Number of n-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b c d e f g h i j Sloane, N. J. A. (ed.). "Sequence A002378 (Oblong (or promic, pronic, or heteromecic) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b c Sloane, N. J. A. (ed.). "Sequence A100827 (Highly cototient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b c d Sloane, N. J. A. (ed.). "Sequence A069099 (Centered heptagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ a b c Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ a b c Sloane, N. J. A. (ed.). "Sequence A080076 (Proth primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ a b c d e f g Sloane, N. J. A. (ed.). "Sequence A001844 (Centered square numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A000957 (Fine's sequence (or Fine numbers): number of relations of valence >= 1 on an n-set; also number of ordered rooted trees with n edges having root of even degree)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b c d e Sloane, N. J. A. (ed.). "Sequence A001106 (9-gonal (or enneagonal or nonagonal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A067128 (Ramanujan's largely composite numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A076980 (Leyland numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A002411 (Pentagonal pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A008918 (Numbers n such that 4*n = (n written backwards))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-14.
- ^ Sloane, N. J. A. (ed.). "Sequence A001190 (Wedderburn-Etherington numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Mackenzie, Dana (2018). "2184: An Absurd (and Adsurd) Tale". Integers. 18.
- ^ Sloane, N. J. A. (ed.). "Sequence A014575 (Vampire numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A082897 (Perfect totient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A001006 (Motzkin numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A005231 (Odd abundant numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A005479 (Prime Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A007629 (Repfigit (REPetitive FIbonacci-like diGIT) numbers (or Keith numbers))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A006886 (Kaprekar numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A005900 (Octahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ a b c Sloane, N. J. A. (ed.). "Sequence A002407 (Cuban primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ a b c d e Sloane, N. J. A. (ed.). "Sequence A006562 (Balanced primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A002110 (Primorial numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ "The Small Groups library". Archived from the original on 2007-02-04. Retrieved 2008-01-22..
- ^ Sloane, N. J. A. (ed.). "Sequence A005898 (Centered cube numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A069151 (Concatenations of consecutive primes, starting with 2, that are also prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A002104 (Logarithmic numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A000129 (Pell numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ a b Sloane, N. J. A. (ed.). "Sequence A002997 (Carmichael numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A000258 (Expansion of e.g.f. exp(exp(exp(x)-1)-1))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A000219 (Number of planar partitions (or plane partitions) of n)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A000931 (Padovan sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A000045 (Fibonacci numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ "Odd numbers that are not of the form x^2+y^2+10*z^2.". The Online Encyclopedia of Integer Sequences. The OEIS Foundation, Inc. Retrieved 13 November 2012.
- ^ Ono, Ken (1997). "Ramanujan, taxicabs, birthdates, zipcodes and twists" (PDF). American Mathematical Monthly. 104 (10): 912–917. CiteSeerX 10.1.1.514.8070. doi:10.2307/2974471. JSTOR 2974471. Archived from the original (PDF) on 15 October 2015. Retrieved 11 November 2012.
- ^ Ono, Ken; K Soundararajan (1997). "Ramanujan's ternary quadratic forms" (PDF). Inventiones Mathematicae. 130 (3): 415–454. Bibcode:1997InMat.130..415O. CiteSeerX 10.1.1.585.8840. doi:10.1007/s002220050191. S2CID 122314044. Archived from the original (PDF) on 18 July 2019. Retrieved 12 November 2012.
- ^ Sloane, N. J. A. (ed.). "Sequence A000979 (Wagstaff primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A001792 (a(n) = (n+2)*2^(n-1))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A144974 (Centered heptagonal prime numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A000078 (Tetranacci numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Pandharipande, Rahul (1998), "Rational curves on hypersurfaces (after A. Givental)", Astérisque, 1997/98 (252): 307–340, arXiv:math/9806133, Bibcode:1998math......6133P, MR 1685628
- ^ Sloane, N. J. A. (ed.). "Sequence A002559 (Markoff (or Markov) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A006958 (Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A001599 (Harmonic or Ore numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A000014 (Number of series-reduced trees with n nodes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Sloane, N. J. A. (ed.). "Sequence A195163 (1000-gonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ^ Sloane, N. J. A. (ed.). "Sequence A038823 (Number of primes between n*1000 and (n+1)*1000)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ^ Stein, William A. (10 February 2017). "The Riemann Hypothesis and The Birch and Swinnerton-Dyer Conjecture". wstein.org. Retrieved 6 February 2021.