400 (number) explained

Number:	400
Divisor:	1, 2, 4, 5, 8, 10, 16, 20, 25, 40, 50, 80, 100, 200, 400
Lang1:	Hebrew
Lang1 Symbol:	ת
Lang2:	Armenian
Lang2 Symbol:	Ն
Lang3:	Babylonian cuneiform
Lang4:	Egyptian hieroglyph
Lang4 Symbol:

400 (four hundred) is the natural number following 399 and preceding 401.

Mathematical properties

400 is the square of 20. 400 is the sum of the powers of 7 from 0 to 3, thus making it a repdigit in base 7 (1111).

A circle is divided into 400 grads, which is equal to 360 degrees and 2π radians. (Degrees and radians are the SI accepted units).

400 is a self number in base 10, since there is no integer that added to the sum of its own digits results in 400. On the other hand, 400 is divisible by the sum of its own base 10 digits, making it a Harshad number.

Other fields

Four hundred is also

• .400 (2 hits out of 5 at-bats) is a numerically significant annual batting average statistic in Major League Baseball, last accomplished by Ted Williams of the Boston Red Sox in 1941.
• The number of days in a Gregorian calendar year changes according to a cycle of exactly 400 years, of which 97 are leap years and 303 are common.
• The Sun is approximately 400 times the size of the Moon but is also approximately 400 times farther away from Earth than the Moon is, thus creating the illusion in which the Sun and the Moon in Earth's sky appear to be of similar size.^[1]
• In gematria 400 is the largest single number that can be represented without using the Sophit forms (see Kaph, Mem, Nun, Pe, and Tzade).

Integers from 401 to 499

400s

401

401 is a prime number, tetranacci number, Chen prime, prime index prime

• Eisenstein prime with no imaginary part
• Sum of seven consecutive primes (43 + 47 + 53 + 59 + 61 + 67 + 71)
• Sum of nine consecutive primes (29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61)
• Mertens function returns 0,
• Member of the Mian–Chowla sequence.

402

402 = 2 × 3 × 67, sphenic number, nontotient, Harshad number, number of graphs with 8 nodes and 9 edges^[2]

• HTTP status code for "Payment Required".
• The area code for Nebraska.

403

403 = 13 × 31, heptagonal number, Mertens function returns 0.

• First number that is the product of an emirp pair.
• HTTP 403, the status code for "Forbidden"
• Also in the name of a retirement plan in the United States, 403(b).
• The area code for southern Alberta.

404

404 = 2² × 101, Mertens function returns 0, nontotient, noncototient, number of integer partitions of 20 with an alternating permutation.

• HTTP status code for "Not Found", perhaps the most famous HTTP status code.
• Section 404 of the Sarbanes–Oxley Act.
• One of the three area codes of the Atlanta calling area.

405

405 = 3⁴ × 5, Mertens function returns 0, Harshad number, pentagonal pyramidal number;

• HTTP status code for "Method Not Allowed".
• Area code for central Oklahoma, including Oklahoma City and surrounding suburbs.
• Interstate 405 is a major, heavily traveled freeway in Southern California, known to the local as "The 405".
• Sum of all numbers in a standard (3x3)x(3x3) Sudoku.

406

406 = 2 × 7 × 29, sphenic number, triangular number, centered nonagonal number, nontotient

• HTTP status code for "Not Acceptable".

• 406 is a poem by John Boyle O'Reilly. It was believed to have been the number of one of O'Reilly's prison cells, and was the number of his first hotel room after he arrived in the United States. Hence the number had a mystical significance to him, as intimated in the poem.
• Peugeot 406 car.
• Area code for all of Montana.

407

407 = 11 × 37,

• Sum of cubes of 4, 0 and 7 (4³ + 0³ + 7³ = 407); narcissistic number
• Sum of three consecutive primes (131 + 137 + 139)
• Mertens function returns 0
• Harshad number
• Lazy caterer number
• HTTP status code for "Proxy Authentication Required"
• Area code for Orlando, Florida
• Colloquial name for the Express Toll Route in Ontario

408

408 = 2³ × 3 × 17

• Sum of four consecutive primes (97 + 101 + 103 + 107)
• Sum of eight consecutive primes (37 + 41 + 43 + 47 + 53 + 59 + 61 + 67)
• Pell number
• Mertens function returns 0
• Octagonal number
• Untouchable number
• Harshad number
• HTTP status code for "Request Timeout"
• Area code for the Silicon Valley

409

409 is a prime number, Chen prime, centered triangular number.

• A family of cleaning products, Formula 409
• An engine known as the Chevrolet 409, a 409 cubic inch W-series V8.
• The song "409" by The Beach Boys, inspired by the above engine
• HTTP status code for "Conflict"
• A Green Day song, "409 in Your Coffeemaker", included on their album 1,039/Smoothed Out Slappy Hours
• The area code for a part of eastern Texas
• Venice has 409 bridges.^[3]
• A Bullet For My Valentine song, "Room 409", from the album The Poison
• Joe Paterno holds the record as the winningest head coach in NCAA FBS with 409 victories.

410s

410

410 = 2 × 5 × 41, sphenic number, sum of six consecutive primes (59 + 61 + 67 + 71 + 73 + 79), nontotient, Harshad number, number of triangle-free graphs on 8 vertices

• HTTP status code for "Gone".
• Area Code 410, a telephone area code for the US State of Maryland, representing portions of the state including the Baltimore metropolitan area and the Eastern Shore.

411

411 = 3 × 137, self number,

• HTTP status code for "Length Required", slang for information (see 4-1-1)
• The number of possible FM broadcasting frequencies between 87.50 and 108.00 MHz in 50 kHz spacing countries

412

412 = 2² × 103, nontotient, noncototient, sum of twelve consecutive primes (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59), 412⁶⁴ + 1 is prime

• HTTP status code for "Precondition Failed"
• Area code for Pittsburgh, Pennsylvania.
• Fictitious Police Code for "Overacting" from "St. George and the Dragonet" - Stan Freeberg

413

413 = 7 × 59, Mertens function returns 0, self number, Blum integer

• HTTP status code for "Request Entity Too Large"
• Area code for Western Massachusetts.
• An important and recurring number in the webcomic Homestuck by Andrew Hussie.

414

414 = 2 × 3² × 23, Mertens function returns 0, nontotient, Harshad number, number of balanced partitions of 31

\sum ^{10} _{n =0} { 414 } ^{n} is prime

• HTTP status code for "Request-URI Too Long"
• Area code for Milwaukee, Wisconsin.
• The 414s, a group of hackers from Milwaukee, Wisconsin.

415

415 = 5 × 83, logarithmic number

• HTTP status code for "Unsupported Media Type"
• 415 Records, a record label
• 415 refers to California Penal Code, section 415, pertaining to public fighting, public disturbance, and public use of offensive words likely to provoke an immediate violent reaction.
• Area code 415, a telephone area code for San Francisco, California

416

416 = 2⁵ × 13, number of independent vertex sets and vertex covers in the 6-sunlet graph^[4]

• HTTP status code for "Requested Range Not Satisfiable"
• 416 is also a nickname for the city of Toronto, based on the area code it used before overlay plans added two more area codes.

417

417 = 3 × 139, Blum integer

• HTTP status code for "Expectation Failed". Also the area code for southwestern Missouri, including Springfield, and Joplin.

418

418 = 2 × 11 × 19; sphenic number, balanced number. It is also the fourth 71-gonal number.^[5]

• The sum of the integers between 13 and 31, inclusive.
• The product of its digits as well as the sum of its prime factors are both 32. Also, 131, a strong concatenation of 13 and 31, is the 32nd prime number (while the 32nd composite number is 48).
• The sum of the 84 digits of the 22nd unique prime in decimal (having a very distinct set of digits than all other known terms in the sequence).^[6]
• The number of Abrahadabra
• Hyper Text Coffee Pot Control Protocol status code for "Teapot" as an April Fools' joke.^{[7] [8]}

419

A prime number, Sophie Germain prime, Chen prime, Eisenstein prime with no imaginary part, highly cototient number, Mertens function returns 0

• Refers to the Nigerian advance fee fraud scheme (after the section of the Nigerian Criminal Code it violates)
• The Area Code for Toledo, OH and other surrounding areas.

420s

420

See main article: 420 (number).

See also: 420 (cannabis culture).

421

• A prime number, sum of five consecutive primes (73 + 79 + 83 + 89 + 97), centered square number, also SMTP code meaning the transmission channel will be closing
• Country calling code for Slovakia

422

422 = 2 × 211, Mertens function returns 0, nontotient, since 422 = 20² + 20 + 2 it is the maximum number of regions into which 21 intersecting circles divide the plane.^[9]

423

423 = 3² × 47, Mertens function returns 0, Harshad number, number of secondary structures of RNA molecules with 10 nucleotides

• Country calling code for Liechtenstein

424

424 = 2³ × 53, sum of ten consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61), Mertens function returns 0, refactorable number, self number

425

425 = 5² × 17, pentagonal number, centered tetrahedral number, sum of three consecutive primes (137 + 139 + 149), Mertens function returns 0, the second number that can be expressed as the sum of two squares in three different ways (425 = 20² + 5² = 19² + 8² = 16² + 13²).

• 425 is an area code in Washington State.

426

426 = 2 × 3 × 71, sphenic number, nontotient, untouchable number

427

427 = 7 × 61, Mertens function returns 0. 427! + 1 is prime.

428

428 = 2² × 107, Mertens function returns 0, nontotient, 428³² + 1 is prime

• , a video game

429

429 = 3 × 11 × 13, sphenic number, Catalan number

430s

430

430 = 2 × 5 × 43, number of primes below 3000, sphenic number, untouchable number

431

A prime number, Sophie Germain prime, sum of seven consecutive primes (47 + 53 + 59 + 61 + 67 + 71 + 73), Chen prime, prime index prime, Eisenstein prime with no imaginary part

• It is also the fourth Leyland prime of the second kind.
• Area code 431 is a telephone area code for Manitoba, Canada.

432

432 = 2⁴ × 3³ = 4² × 3³, the sum of four consecutive primes (103 + 107 + 109 + 113), a Harshad number, a highly totient number, an Achilles number and the sum of totient function for first 37 integers. 432! is the first factorial that is not a Harshad number in base 10. 432 is also three-dozen sets of a dozen, making it three gross. An equilateral triangle whose area and perimeter are equal, has an area (and perimeter) equal to

\sqrt{ 432 } .

433

A prime number, Markov number, star number.

• The perfect score in the game show Fifteen To One, only ever achieved once in over 2000 shows.
• 433 can refer to composer John Cage's composition 4′33″ (pronounced "Four minutes, thirty-three seconds" or just "Four thirty-three").

434

434 = 2 × 7 × 31, sphenic number, sum of six consecutive primes (61 + 67 + 71 + 73 + 79 + 83), nontotient, maximal number of pieces that can be obtained by cutting an annulus with 28 cuts^[10]

435

435 = 3 × 5 × 29, sphenic number, triangular number, hexagonal number, self number, number of compositions of 16 into distinct parts

• The number of members in the US House of Representatives.

436

436 = 2² × 109, nontotient, noncototient, lazy caterer number

437

437 = 19 × 23, Blum integer

438

438 = 2 × 3 × 73, sphenic number, Smith number.

• The "438 match" or "438 game" has been used by cricket media to describe the famous 2006 One Day International in which Australia scored a world record 434 in their innings, only to see South Africa respond in their innings with 438.

439

A prime number, sum of three consecutive primes (139 + 149 + 151), sum of nine consecutive primes (31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67), strictly non-palindromic number

440s

440

See main article: 440 (number).

441

441 = 3² × 7² = 21²

• 441 is the sum of the cubes of the first 6 natural numbers (441 = 1³ + 2³ + 3³ + 4³ + 5³ + 6³).
• 441 is a centered octagonal number,^[11] a refactorable number, and a Harshad number.
• 441 is the number of squares on a Super Scrabble board.

442

442 = 2 × 13 × 17 = 21² + 1,^[12] sphenic number, sum of eight consecutive primes (41 + 43 + 47 + 53 + 59 + 61 + 67 + 71)

443

A prime number, Sophie Germain prime, Chen prime, Eisenstein prime with no imaginary part, Mertens function sets new low of -9, which stands until 659.

• In computing, it is the default port for HTTPS connections.

444

444 = 2² × 3 × 37, refactorable number, Harshad number, number of noniamonds without holes, and a repdigit.

• The title of the final track of Autechre's 1993 debut album Incunabula.
• The 444th Fighter Squadron "Spare", a fictional air squadron in .

445

445 = 5 × 89, number of series-reduced trees with 17 nodes

446

446 = 2 × 223, nontotient, self number

447

447 = 3 × 149, number of 1's in all partitions of 22 into odd parts

• The flight number of Air France Flight 447

448

448 = 2⁶ × 7, untouchable number, refactorable number, Harshad number

449

A prime number, sum of five consecutive primes (79 + 83 + 89 + 97 + 101), Chen prime, Eisenstein prime with no imaginary part, Proth prime. Also the largest number whose factorial is less than 10¹⁰⁰⁰

450s

450

450 = 2 × 3² × 5², nontotient, sum of totient function for first 38 integers, refactorable number, Harshad number,

• SMTP code meaning the requested mail action was not carried out.
• A perfect score in Canadian five-pin bowling.
• An area code in Southern Quebec.

451

451 = 11 × 41; 451 is a Wedderburn–Etherington number and a centered decagonal number; its reciprocal has period 10; 451 is the smallest number with this period reciprocal length.

• The novel Fahrenheit 451 refers to the temperature in Fahrenheit that author Ray Bradbury understood to be the autoignition point of paper.
  • By extension, the numbers "451" are often included as the first security code a player encounters in Immersive sim video games as a reference to the System Shock series of games which first included the code as their own reference to Bradbury's novel.^[13]
• HTTP status code for "Unavailable For Legal Reasons" a HTTP response error when the user requests an illegal resource, such as a web page censored by a government.^[14]

452

452 = 2² × 113, number of surface-points of a tetrahedron with edge-length 15

• SMTP code meaning that the requested mail action was not carried out because of insufficient system storage

453

453 = 3 × 151, Blum integer

454

454 = 2 × 227, nontotient, a Smith number

455

455 = 5 × 7 × 13, sphenic number, tetrahedral number

• 455 Rocket is the title of a song by Kathy Mattea
• 455 kHz is a standard intermediate frequency for analog superheterodyne AM broadcast band receivers.
• The sum of the squares of the first 455 primes is divisible by 455.

456

456 = 2³ × 3 × 19, sum of a twin prime (227 + 229), sum of four consecutive primes (107 + 109 + 113 + 127), centered pentagonal number, icosahedral number

• In the TV show, the antagonists were an alien species with the designation 456.
• Number of contestants in the South Korean Netflix drama Squid Game.

457

• A prime number, sum of three consecutive primes (149 + 151 + 157), self number.
• The international standard frequency for radio avalanche transceivers (457 kHz).

458

458 = 2 × 229, nontotient, number of partitions of 24 into divisors of 24

459

459 = 3³ × 17, triangular matchstick number^[15]

• 459 West 18th Street, a residential building at that address in Manhattan's West Chelsea neighborhood, built in 2008.

460s

460

460 = 2² × 5 × 23, centered triangular number, dodecagonal number, Harshad number, sum of twelve consecutive primes (17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61)

461

A prime number, Chen prime, sexy prime with 467, Eisenstein prime with no imaginary part, prime index prime

462

\tbinom { 11 } 5 , stirling number of the second kind \left\{ { 9 \atop 7 } \right\} , sum of six consecutive primes (67 + 71 + 73 + 79 + 83 + 89), pronic number, sparsely totient number, idoneal number

463

A prime number, sum of seven consecutive primes (53 + 59 + 61 + 67 + 71 + 73 + 79), centered heptagonal number. This number is the first of seven consecutive primes that are one less than a multiple of 4 (from 463 to 503).

• Number of days in the synodic period of Ceres
• A common baseball double play (see baseball positions)
• A single by Buck 65, named after the baseball term

464

See also: 4-6-4. 464 = 2⁴ × 29, primitive abundant number, since 464 = 21² + 21 + 2 it is the maximum number of regions into which 22 intersecting circles divide the plane,^[9] maximal number of pieces that can be obtained by cutting an annulus with 29 cuts^[10]

• In chess it is the number of legal positions of the kings, not counting mirrored positions. Has some importance when constructing an endgame tablebase.
• Model number of the home computer Amstrad CPC 464.

465

465 = 3 × 5 × 31, sphenic number, triangular number, member of the Padovan sequence, Harshad number

466

466 = 2 × 233, noncototient, lazy caterer number.

467

A prime number, safe prime, sexy prime with 461, Chen prime, Eisenstein prime with no imaginary part

\sum ^{10} _{n =0} { 467 } ^{n} is prime

468

468 = 2² × 3² × 13, sum of ten consecutive primes (29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67), refactorable number, self number, Harshad number

469

469 = 7 × 67, centered hexagonal number. 469! - 1 is prime.

470s

470

470 = 2 × 5 × 47, sphenic number, nontotient, noncototient, cake number

• In golf, 470 is the minimum length in yards from the tee to the hole on a Par 5.
• 470 is an Olympic class of sailing dinghy

471

471 = 3 × 157, sum of three consecutive primes (151 + 157 + 163), perfect totient number, φ(471) = φ(σ(471)).^[16]

472

472 = 2³ × 59, nontotient, untouchable number, refactorable number, number of distinct ways to cut a 5 × 5 square into squares with integer sides

• The Amstrad CPC472 was a short-lived home computer for the Spanish market.

473

473 = 11 × 43, sum of five consecutive primes (83 + 89 + 97 + 101 + 103), Blum integer

474

474 = 2 × 3 × 79, sphenic number, sum of eight consecutive primes (43 + 47 + 53 + 59 + 61 + 67 + 71 + 73), nontotient, noncototient, sum of totient function for first 39 integers, untouchable number, nonagonal number

475

475 = 5² × 19, 49-gonal number, member of the Mian–Chowla sequence.

476

476 = 2² × 7 × 17, Harshad number, admirable number

477

477 = 3² × 53, pentagonal number

478

478 = 2 × 239, Companion Pell number, number of partitions of 26 that do not contain 1 as a part

479

A prime number, safe prime, sum of nine consecutive primes (37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71), Chen prime, Eisenstein prime with no imaginary part, self number

• Also an area code in the U.S. state of Arkansas.

480s

480

480 = 2⁵ × 3 × 5, sum of a twin prime (239 + 241), sum of four consecutive primes (109 + 113 + 127 + 131), highly totient number, refactorable number, Harshad number

\sum ^{10} _{n =0} { 480 } ^{n} is prime

481

481 = 13 × 37, octagonal number, centered square number, Harshad number

482

482 = 2 × 241, nontotient, noncototient, number of series-reduced planted trees with 15 nodes

483

483 = 3 × 7 × 23, sphenic number, Smith number

484

484 = 2² × 11² = 22², palindromic square, nontotient

485

485 = 5 × 97, number of triangles (of all sizes, including holes) in Sierpiński's triangle after 5 inscriptions^[17]

• RS-485

486

486 = 2 × 3⁵, Harshad number, Perrin number

• Shorthand for the Intel 80486 microprocessor chip.

487

A prime number, sum of three consecutive primes (157 + 163 + 167), Chen prime,

• The only primes under 7.74 × 10¹³ that divide their own decimal repetends are 3, 487, and 56598313.^[18]
• Shorthand for the Intel 80487 floating point processor chip.

488

488 = 2³ × 61, nontotient, refactorable number, φ(488) = φ(σ(488)),^[16] number of surface points on a cube with edge-length 10.^[19]

489

489 = 3 × 163, octahedral number

490s

490

490 = 2 × 5 × 7², noncototient, sum of totient function for first 40 integers, number of integer partitions of 19,^[20] self number.

• An (possibly arbitrary) large number in the Christian Gospel of Matthew. In Matthew 18:21–35, Jesus tells the Parable of the Unforgiving Servant, instructing Peter to forgive his brother "seventy times seven" times when his brother sins against him http://www.biblegateway.com/passage/?search=Matthew+18:21-35.

491

A prime number, isolated prime, Sophie Germain prime, Chen prime, Eisenstein prime with no imaginary part, strictly non-palindromic number

492

492 = 2² × 3 × 41, sum of six consecutive primes (71 + 73 + 79 + 83 + 89 + 97), refactorable number, member of a Ruth–Aaron pair with 493 under first definition

493

493 = 17 × 29, sum of seven consecutive primes (59 + 61 + 67 + 71 + 73 + 79 + 83), member of a Ruth–Aaron pair with 492 under first definition, the 493d centered octagonal number is also a centered square number^[21]

494

494 = 2 × 13 × 19 =

\left\langle \left\langle { 8 \atop 1 } \right\rangle \right\rangle ,^[22] sphenic number, nontotient

495

See main article: 495 (number).

496

See main article: 496 (number).

497

497 = 7 × 71, sum of five consecutive primes (89 + 97 + 101 + 103 + 107), lazy caterer number.

498

498 = 2 × 3 × 83, sphenic number, untouchable number, admirable number, abundant number

499

A prime number, isolated prime, Chen prime, 4⁴⁹⁹ - 3⁴⁹⁹ is prime

Notes and References

1. Web site: 2013-06-26 . Why do the sun and moon seem like the same size? Space EarthSky . 2022-10-28 . earthsky.org . en-US.
2. Triangle T(n,k) read by rows, giving number of graphs with n nodes (n >= 1) and k edges (0 <= k <= n(n-1)/2).
3. Web site: Venice: The City Built on Water . 2022-09-21 . Google Maps.
4. a(n) = 2*a(n-1) + 2*a(n-2) for n > 1; a(0)=2, a(1)=2.
5. Book: Conway . John H. . John Horton Conway . Guy . Richard . Richard K. Guy . The Book of Numbers . 2012 . . 39 . 10.1007/978-1-4612-4072-3 . 978-1-4612-4072-3 . 39220031 . en .
6. 2022-05-20.

   That number is 142,857,157,142,857,142,856,999,999,985,714,285,714,285,857,142,857,142,855,714,285,571,428,571,428,572,857,143.

7. Hyper Text Coffee Pot Control Protocol (HTCPCP/1.0). RFC. Any attempt to brew coffee with a teapot should result in the error code "418 I'm a teapot". The resulting entity body MAY be short and stout.. 13 Sep 2018. 1 April 1998. L. Masinter. Network Working Group. 10.17487/RFC2324 .
8. The Hyper Text Coffee Pot Control Protocol for Tea Efflux Appliances (HTCPCP-TEA). TEA-capable pots that are not provisioned to brew coffee may return either a status code of 503, indicating temporary unavailability of coffee, or a code of 418 as defined in the base HTCPCP specification to denote a more permanent indication that the pot is a teapot.. RFC. 13 Sep 2018. 2070-1721. 1 April 2014. I. Nazar. IETF Request for Comments (RFC) Pages - Test. 10.17487/RFC7168 .
9. a(n) = n^2 + n + 2.
10. a(n) = n*(n+3)/2.
11. Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers.
12. a(n) = n^2 + 1.
13. Web site: LeBlanc . Marc . OG System Shock dev plays remake 1 . YouTube . 18 August 2023.
14. Web site: 451 Unavailable For Legal Reasons - HTTP MDN. 2021-04-23. developer.mozilla.org.
15. Triangular matchstick numbers: a(n) = 3*n*(n+1)/2.
16. Numbers k such that phi(k) = phi(sigma(k)).
17. a(0)=1, a(n) = 3*a(n-1) + 2; a(n) = 2*3^n - 1.
18. Primes p such that 10^(p-1)

    1 (mod p^2)

    .
19. a(n) = 6*n^2 + 2 for n > 0, a(0)=1.
20. a(n) = number of partitions of n (the partition numbers).
21. a(n) = 6*a(n-1) - a(n-2) - 2 with a(0) = 1, a(1) = 3.
22. Second-order Eulerian triangle T(n, k), 1 <= k <= n.