2000 (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 | |||
| Quaternary | 1331004 | |||
| Quinary | 310005 | |||
| Senary | 131326 | |||
| Octal | 37208 | |||
| Duodecimal | 11A812 | |||
| Hexadecimal | 7D016 | |||
| Vigesimal | 50020 | |||
| Base 36 | 1JK36 | |||
| Egyptian hieroglyph | 𓆽 | |||
2000 (two thousand) is a natural number after 1999 and before 2001.[1] It also appears in the calendar year.
In Roman Numerals, 2000 is written as MM.[1]
In mathematics
[change | change source]The number 2000 is:
- the highest number expressible using only two unmodified characters in Roman numerals (MM)
- an regular number[1]
- an Achilles number[2]
- the smallest four digit eban number[3]
- the sum of all the nban numbers.[4]
Factors
[change | change source]The factors of 2000 are: 1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 125, 200, 250, 400, 500, 1000 and 2000.[1]
Multiples
[change | change source]Here are the first 10 multiples of 2000:
| Multiplication | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| 2000 * x | 2000 | 4000 | 6000 | 8000 | 100000 | 12000 | 14000 | 16000 | 18000 | 20000 |
And the 20 multiples:
| Multiplication | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|
| 2000 * x | 22000 | 24000 | 26000 | 28000 | 300000 | 32000 | 34000 | 36000 | 38000 | 40000 |
Numbers in the range 2001–2999
[change | change source]These are chosen numbers in the range of 2001–2999.
2001 to 2099
[change | change source]2002
[change | change source]2002 = 2 × 7 × 11 × 13. It is a palindromic number in decimal, base 76, 90, 142, and 11 other non-trivial bases.
2003
[change | change source]2003 is a Sophie Germain prime.
2005
[change | change source]2005 = 5 × 401. It is a vertically symmetric number.
2011
[change | change source]2011 is a sexy prime with 2017. It is also the sum of eleven consecutive primes: 2011 = 157 + 163 + 167 + 173 + 179 + 181 + 191 + 193 + 197 + 199 + 211
2015
[change | change source]2015 = 5 × 13 × 31. It is a Lucas–Carmichael number.[5]
2016
[change | change source]2016 = 25 × 32 × 7. It is the second-smallest Erdős–Nicolas number[6]and a triangular number.[7]
2017
[change | change source]2017 is a sexy prime with 2011.
2024
[change | change source]2024 = 23 × 11 × 23. It is a tetrahedral number.[8]
2025
[change | change source]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,[9] lowest number with exactly 15 odd divisors.[10] Sum of odd numbers from 1 to 89.
2027
[change | change source]2027 is a super-prime and a safe prime.[11]
2029
[change | change source]2029 is a prime number and a member of the Mian–Chowla sequence.[12]
2031
[change | change source]2031 = 3 × 677. It is a centered pentagonal number.[13]
2035
[change | change source]2035 = 5 × 11 × 37. It is a Wolstenholme number.[14]
2036
[change | change source]2036 = 22 × 509. It is an Eulerian number.[15]
2039
[change | change source]2039 is a Sophie Germain prime and a safe prime.[11]
2047
[change | change source]2047 = 23 × 89 = 211 − 1 and it is the first Mersenne number that is composite for a prime exponent. It is a super-Poulet number,[16] a Woodall number,[17] a decagonal number,[18] and a centered octahedral number.[19]
2053
[change | change source]2053 is a prime number and a star number.
2056
[change | change source]2056 = 23 × 257. It is the magic constant of an n × n normal magic square and the n-queens problem for n = 16.
2063
[change | change source]2063 is a Sophie Germain prime, a safe prime,[11] and a super-prime.
2066
[change | change source]2066 = 2 × 1033. It is a Bell number.[20]
2069
[change | change source]2069 is a Sophie Germain prime.
2080
[change | change source]2080 = 25 × 5 × 13. It is a triangular number.
2081, 2083, 2087, and 2089
[change | change source]2081, 2083, 2087, and 2089 form a prime quadruplet.
2081 is also a super-prime.
2099
[change | change source]2099 is a super-prime, a safe prime,[11] and a highly cototient number.[21]
2100 to 2199
[change | change source]2101
[change | change source]2101 = 11 × 191. It is a centered heptagonal number.[22]
2107 and 2108
[change | change source]2107 and 2108 form a Ruth–Aaron pair under the first definition.
2109
[change | change source]2109 = 3 × 19 × 37. It is a square pyramidal number.[23]
2113
[change | change source]2113 is a Proth prime[24] and a centered square number.[25]
2125
[change | change source]2125 = 53 × 17. It is a nonagonal number.[26]
2129
[change | change source]2129 is a Sophie Germain prime.
2141
[change | change source]2141 is a Sophie Germain prime.
2145
[change | change source]2145 = 3 × 5 × 11 × 13. It is a triangular number.
2160
[change | change source]2160 = 24 × 33 × 5. It is a largely composite number.[27]
2176
[change | change source]2176 = 27 × 17. It is a pentagonal pyramidal number[28] and a centered pentagonal number.[13]
2179
[change | change source]2179 is a prime number and a Wedderburn–Etherington number.[29]
2187
[change | change source]2187 = 37. It is a vampire number[30] and a perfect totient number.[31]
2188
[change | change source]2188 = 22 × 547. It is a Motzkin number.[32]
2199
[change | change source]2199 = 3 × 733. It is a perfect totient number.[31]
2200 to 2299
[change | change source]2201
[change | change source]2201 – only known non-palindromic number whose cube is palindromic; also no known fourth or higher powers are palindromic for non-palindromic numbers
2207
[change | change source]2207 – safe prime,[11] Lucas prime[33]
2208
[change | change source]2208 – Keith number[34]
2209
[change | change source]2209 = 472, palindromic in base 14 (B3B14), centered octagonal number[9]
2211
[change | change source]2211 – triangular number
2221
[change | change source]2221 – super-prime, happy number
2223
[change | change source]2223 – Kaprekar number[35]
2232
[change | change source]2232 – decagonal number[18]
2245
[change | change source]2245 – centered square number[25]
2254
[change | change source]2254 – member of the Mian–Chowla sequence[12]
2255
[change | change source]2255 – octahedral number[36]
2269
[change | change source]2269 – super-prime, cuban prime[37]
2273
[change | change source]2273 – Sophie Germain prime
2276
[change | change source]2276 _ centered heptagonal number[22]
2278
[change | change source]2278 – triangular number
2281
[change | change source]2281 – star number,
2287
[change | change source]2287 – balanced prime[38]
2299
[change | change source]2299 – member of a Ruth–Aaron pair with 2300 (first definition)
2300 to 2399
[change | change source]2300
[change | change source]2300 – tetrahedral number,[8] member of a Ruth–Aaron pair with 2299 (first definition)
2301
[change | change source]2301 – nonagonal number[26]
2309
[change | change source]2309 – primorial prime, twin prime with 2311, Mertens function zero, highly cototient number[21]
2310
[change | change source]2311
[change | change source]2311 – primorial prime, twin prime with 2309
2326
[change | change source]2326 – centered pentagonal number[13]
2328
[change | change source]2328 – sum of the totient function for the first 87 integers, the number of groups of order 128[40]
2331
[change | change source]2331 – centered cube number[41]
2339
[change | change source]2339 – Sophie Germain prime, twin prime with 2341
2341
[change | change source]2341 – super-prime, twin prime with 2339
2346
[change | change source]2346 – triangular number
2351
[change | change source]2351 – Sophie Germain prime, super-prime
2352
[change | change source]2352 – pronic number[42]
2357
[change | change source]2357 – Smarandache–Wellin prime[43]
2378
[change | change source]2378 – Pell number[44]
2379
[change | change source]2379 – member of the Mian–Chowla sequence[12]
2381
[change | change source]2381 – super-prime, centered square number[25]
2393
[change | change source]2393 – Sophie Germain prime
2399
[change | change source]2399 – Sophie Germain prime
2400 to 2499
[change | change source]- 2401 = 492 = 74, centered octagonal number[9]
- 2415 – triangular number
- 2417 – super-prime, balanced prime[38]
- 2425 – decagonal number[18]
- 2437 – cuban prime,[37] largest right-truncatable prime in base 5
- 2447 – safe prime[11]
- 2450 – pronic number[42]
- 2458 – centered heptagonal number[22]
- 2459 – Sophie Germain prime, safe prime[11]
- 2465 – magic constant of n × n normal magic square and n-queens problem for n = 17, Carmichael number[45]
- 2470 – square pyramidal number[23]
- 2477 – super-prime, cousin prime
- 2481 – centered pentagonal number[13]
- 2484 – nonagonal number[26]
- 2485 – triangular number, number of planar partitions of 13[46]
- 2491 = member of Ruth–Aaron pair with 2492 under second definition
- 2492 – member of Ruth–Aaron pair with 2491 under second definition
2500 to 2599
[change | change source]- 2510 – member of the Mian–Chowla sequence[12]
- 2513 – member of the Padovan sequence[47]
- 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[25]
- 2530 – Leyland number[48]
- 2543 – Sophie Germain prime, sexy prime with 2549
- 2549 – Sophie Germain prime, super-prime, sexy prime with 2543
- 2550 – pronic number[42]
- 2556 – triangular number
- 2579 – safe prime,[11] palindromic prime in bases 5, 25, and 28: 2579 = 403045 = 43425 = 38328
- 2580 – Keith number,[34]
- 2584 – Fibonacci number,[49]
2600 to 2699
[change | change source]- 2600 – tetrahedral number,[8] member of a Ruth–Aaron pair with 2601 (first definition)
- 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[19]
- 2626 – decagonal number[18]
- 2628 – triangular number
- 2641 – centered pentagonal number[13]
- 2647 – super-prime, centered heptagonal number[22]
- 2652 – pronic number[42]
- 2665 – centered square number[25]
- 2674 – nonagonal number[26]
- 2677 – balanced prime[38]
- 2680 – number of 11-queens problem solutions
- 2683 – super-prime
- 2689 – Proth prime[24]
- 2693 – Sophie Germain prime
- 2699 – Sophie Germain prime
2700 to 2799
[change | change source]- 2701 – triangular number, super-Poulet number[16]
- 2702 – sum of the totient function for the first 94 integers
- 2707 – strong prime
- 2719 – super-prime, largest known odd number which cannot be expressed in the form x2 + y2 + 10z2 where x, y and z are integers.[50] In 1997 it was conjectured that this is also the largest such odd number.[51] It is now[when?] known this is true if the generalized Riemann hypothesis is true.[52]
- 2728 – Kaprekar number[35]
- 2729 – highly cototient number[21]
- 2731 – the only Wagstaff prime with four digits,[53] Jacobsthal prime
- 2736 – octahedral number[36]
- 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[24]
- 2756 – pronic number[42]
- 2774 – sum of the totient function for the first 95 integers
- 2775 – triangular number
- 2780 – member of the Mian–Chowla sequence[12]
- 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[37]
2800 to 2899
[change | change source]- 2801 – first base 7 repunit prime
- 2803 – super-prime
- 2806 – centered pentagonal number,[13] sum of the totient function for the first 96 integers
- 2809 = 532, centered octagonal number[9]
- 2813 – centered square number[25]
- 2819 – Sophie Germain prime, safe prime, sum of seven consecutive primes (383 + 389 + 397 + 401 + 409 + 419 + 421)[11]
- 2821 – Carmichael number[45]
- 2835 – decagonal number[18]
- 2843 – centered heptagonal prime[54]
- 2850 – triangular number
- 2862 – pronic number[42]
- 2870 – square pyramidal number[23]
- 2871 – nonagonal number[26]
- 2872 – tetranacci number[55]
- 2875 – number of lines on a quintic threefold[56]
- 2879 – safe prime[11]
- 2897 – super-prime, Markov prime[57]
2900 to 2999
[change | change source]- 2903 – Sophie Germain prime, safe prime,[11] balanced prime[38]
- 2909 – super-prime
- 2915 – Lucas–Carmichael number[5]
- 2924 – amicable number with 2620
- 2925 – magic constant of n × n normal magic square and n-queens problem for n = 18, tetrahedral number,[8] member of the Mian-Chowla sequence[12]
- 2926 – triangular number
- 2939 – Sophie Germain prime
- 2963 – Sophie Germain prime, safe prime, balanced prime[38]
- 2965 – greater of second pair of Smith brothers, centered square number[25]
- 2969 – Sophie Germain prime
- 2970 – harmonic divisor number,[58] pronic number[42]
- 2976 – centered pentagonal number[13]
- 2999 – safe prime
Prime numbers
[change | change source]There are 127 prime numbers between 2000 and 3000:[59][60]
- 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
[change | change source]- 1 2 3 4 Vanovschi, Vitalii. "Properties of the number 2000". www.numberempire.com. Archived from the original on 2022-12-31. Retrieved 2026-09-24.
- ↑ 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.
- 1 2 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.
- ↑ Sloane, N. J. A. (ed.). "Sequence A000217 (Triangular numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 4 Sloane, N. J. A. (ed.). "Sequence A000292 (Tetrahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 4 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.
- 1 2 3 4 5 6 7 8 9 10 11 Sloane, N. J. A. (ed.). "Sequence A005385 (Safe primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 4 5 6 Sloane, N. J. A. (ed.). "Sequence A005282 (Mian-Chowla sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 4 5 6 7 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 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.
- 1 2 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.
- 1 2 3 4 5 Sloane, N. J. A. (ed.). "Sequence A001107 (10-gonal (or decagonal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 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 A011971 (Aitken's array)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 Sloane, N. J. A. (ed.). "Sequence A100827 (Highly cototient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- 1 2 3 4 Sloane, N. J. A. (ed.). "Sequence A069099 (Centered heptagonal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 Sloane, N. J. A. (ed.). "Sequence A000330 (Square pyramidal numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 Sloane, N. J. A. (ed.). "Sequence A080076 (Proth primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 4 5 6 7 Sloane, N. J. A. (ed.). "Sequence A001844 (Centered square numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 4 5 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.
- ↑ 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 A001190 (Wedderburn-Etherington numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- ↑ Sloane, N. J. A. (ed.). "Sequence A014575 (Vampire numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 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.
- ↑ Sloane, N. J. A. (ed.). "Sequence A005479 (Prime Lucas numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 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.
- 1 2 Sloane, N. J. A. (ed.). "Sequence A006886 (Kaprekar numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 Sloane, N. J. A. (ed.). "Sequence A005900 (Octahedral numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 Sloane, N. J. A. (ed.). "Sequence A002407 (Cuban primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 3 4 5 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.
- 1 2 3 4 5 6 7 Sloane, N. J. A. (ed.). "Sequence A002378 (Oblong (or promic, pronic, or heteromecic) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
- ↑ 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 A000129 (Pell numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-13.
- 1 2 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 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 A076980 (Leyland numbers)". 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. 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. 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 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 A001599 (Harmonic or Ore 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.
