400 (number)

401

401 is a prime number, a Chen prime,^[1] a prime index prime, an Eisenstein prime with no imaginary part, a tetranacci number,^[2] a zero of Mertens function,^[3] and a member of the Mian–Chowla sequence.^[4]

It is the sum of seven consecutive primes (43 + 47 + 53 + 59 + 61 + 67 + 71) and the sum of nine consecutive primes (29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61).

402

402 = 2 × 3 × 67. It is a sphenic number, a nontotient and a Harshad number. There are 402 graphs with 8 nodes and 9 edges.^[5]

403

403 = 13 × 31. It is a heptagonal number, a zero of Mertens function,^[3] and the smallest number that is the product of an emirp pair.^[6]

404

404 = 2² × 101, a zero of Mertens function,^[3] a nontotient and a noncototient. There are 404 integer partitions of 20 with an alternating permutation.^[7]

The HTTP 404 status code is usually sent from a web page if a user attempts to reach a broken or dead link. It has since become one of the most commonly reached, and thus most recognizable errors on the World Wide Web.^[8]

405

405 = 3⁴ × 5. It is a zero of Mertens function,^[3] a Harshad number and a pentagonal pyramidal number.

406

406 = 2 × 7 × 29. It is a sphenic number, the 28th triangular number,^[9] a centered nonagonal number,^[10] an even nontotient, and a Narayana's cow number.^[11]

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406

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.

407

407 = 11 × 37. It is a zero of Mertens function,^[3] a Harshad number, and a Lazy caterer number.^[12]It is the sum of the cubes of 4, 0 and 7 (4³ + 0³ + 7³ = 407) and thus a narcissistic number.^[13] It is the sum of three consecutive primes (131 + 137 + 139).

408

408 = 2³ × 3 × 17. It is a Pell number,^[14] a zero of Mertens function,^[3] an Octagonal number,^[15] an Untouchable number^[16] and a Harshad number. It is the sum of four consecutive primes (97 + 101 + 103 + 107) and the sum of eight consecutive primes (37 + 41 + 43 + 47 + 53 + 59 + 61 + 67).

409

409 is a prime number, a Chen prime,^[1] and a centered triangular number.^[17]

410

410 = 2 × 5 × 41. It is a sphenic number, a nontotient, a Harshad number, and the sum of six consecutive primes (59 + 61 + 67 + 71 + 73 + 79). There are 410 triangle-free graphs on 8 vertices.^[18]

411

411 = 3 × 137. It is a self number.^[19]

412

412 = 2² × 103. It is a nontotient, a noncototient, and the sum of twelve consecutive primes (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59).

412⁶⁴ + 1 is prime

413

413 = 7 × 59. It is a zero of Mertens function,^[3] a self number,^[19] and a Blum integer.

414

414 = 2 × 3² × 23. It is a zero of Mertens function,^[3] a nontotient, and a Harshad number. There are 414 balanced partitions of 31.^[20]

${\displaystyle \sum _{n=0}^{10}{414}^{n}}$ is prime^[21]

415

415 = 5 × 83. It is a logarithmic number.^[22]

416

416 = 2⁵ × 13. There are independent vertex sets and vertex covers in the 6-sunlet graph^[23]

417

417 = 3 × 139. It is a Blum integer.

418

418 = 2 × 11 × 19. It is a sphenic number,^[24] a balanced number.^[25] and the fourth 71-gonal number.^[26]

It is the sum of the integers between 13 and 31, inclusive.

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).^[27]

419

419 is a prime number, a Sophie Germain prime,^[28] a Chen prime,^[1] an Eisenstein prime with no imaginary part, a highly cototient number,^[29] and a zero of Mertens function.^[3]

420

421

421 is a prime number, a centered square number,^[30] and the sum of five consecutive primes (73 + 79 + 83 + 89 + 97).

422

422 = 2 × 211. It is a zero of Mertens function^[3] and a nontotient. Since 422 = 20² + 20 + 2, 422 is the maximum number of regions into which 21 intersecting circles divide the plane.^[31]

423

423 = 3² × 47. It is a zero of Mertens function^[3] and a Harshad number.

There are 423 secondary structures of RNA molecules with 10 nucleotides^[32]

424

424 = 2³ × 53. It is a zero of Mertens function,^[3] a refactorable number,^[33] a self number^[19] and the sum of ten consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61).

425

425 = 5² × 17. It is a pentagonal number,^[34] a centered tetrahedral number, a zero of Mertens function,^[3] and the sum of three consecutive primes (137 + 139 + 149). It is the second number that can be expressed as the sum of two squares in three different ways (425 = 20² + 5² = 19² + 8² = 16² + 13²).

426

426 = 2 × 3 × 71. It is a sphenic number, a nontotient, and an untouchable number.

427

427 = 7 × 61. It is a zero of Mertens function.^[3]

427! + 1 is prime.

428

428 = 2² × 107. It is a zero of Mertens function and a nontotient.

428³² + 1 is prime^[35]

429

429 = 3 × 11 × 13. It is a sphenic number and a Catalan number.^[36]

430

430 = 2 × 5 × 43. It is a sphenic number and an untouchable number.^[16]

431

431 is a prime number, a Sophie Germain prime,^[28] a Chen prime,^[1] a prime index prime, an Eisenstein prime with no imaginary part, the fourth Leyland prime of the second kind and the sum of seven consecutive primes (47 + 53 + 59 + 61 + 67 + 71 + 73).

432

432 = 2⁴ × 3³ = 4² × 3³. It is a Harshad number, a highly totient number,^[37] an Achilles number and the sum of the totient function for first 37 integers. It is the sum of four consecutive primes (103 + 107 + 109 + 113). 432! is the first factorial that is not a Harshad number in base 10. An equilateral triangle whose area and perimeter are equal has an area (and perimeter) equal to ${\displaystyle {\sqrt {432}}}$.

433

433 is a prime number, a Markov number,^[38] and a star number.^[39]

434

434 = 2 × 7 × 31. It is a sphenic number, a nontotient and the sum of six consecutive primes (61 + 67 + 71 + 73 + 79 + 83). It is the maximal number of pieces that can be obtained by cutting an annulus with 28 cuts.^[40]

435

435 = 3 × 5 × 29. It is a sphenic number, a hexagonal number,^[41] a self number,^[19] and the 29th triangular number.^[42] There are 435 compositions of 16 into distinct parts.^[43]

436

436 = 2² × 109. It is a nontotient, a noncototient, and a lazy caterer number.^[12]

437

437 = 19 × 23. It is a Blum integer.

438

438 = 2 × 3 × 73. It is a sphenic number and a Smith number.^[44]

439

439 is a prime number and a strictly non-palindromic number.^[45] It is the sum of three consecutive primes (139 + 149 + 151) and the sum of nine consecutive primes (31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67).

440

440 = 2³ × 5 × 11. It is a Harshad number, an abundant number,^[46] a happy number,^[47] and the sum of the first 17 prime numbers.

A440 (pitch standard) is widely used for the musical note A4, the A above Middle C, when tuning instruments.

441

441 = 3² × 7² = 21. It is a centered octagonal number,^[48] a refactorable number,^[33] and a Harshad number.

441 is also the sum of the cubes of the first 6 natural numbers: 441 = 1³ + 2³ + 3³ + 4³ + 5³ + 6³.

442

442 = 2 × 13 × 17. It is a sphenic number and the sum of eight consecutive primes (41 + 43 + 47 + 53 + 59 + 61 + 67 + 71).

443

443 is a prime number, a Sophie Germain prime,^[28] a Chen prime,^[1] and an Eisenstein prime with no imaginary part.

Mertens function of 443 is -9, a new low which stands until 659.

444

444 = 2² × 3 × 37. It is a refactorable number,^[33] a Harshad number, and a repdigit. There are 444noniamonds without holes.^[49]

445

445 = 5 × 89. There are 445 series-reduced trees with 17 nodes.^[50]

446

446 = 2 × 223. It is a nontotient and a self number.^[19]

447

447 = 3 × 149. There are 447 1's in all partitions of 22 into odd parts.^[51]

448

448 = 2⁶ × 7. It is an untouchable number,^[16] a refactorable number,^[33] and a Harshad number.

449

449 is a prime number, a Chen prime,^[1] a Proth prime,^[52] an Eisenstein prime with no imaginary part, and the sum of five consecutive primes (79 + 83 + 89 + 97 + 101).

449! is the largest factorial less than 10¹⁰⁰⁰.

450

450 = 2 × 3² × 5². It is a nontotient, a refactorable number,^[33] a Harshad number, and the sum of totient function for the first 38 integers.

451

451 = 11 × 41. It is a Wedderburn–Etherington number^[53] and a centered decagonal number.^[54] It is the smallest number whose reciprocal has period 10.

452

452 = 2² × 113. There are 452 surface-points of a tetrahedron with edge-length 15.^[55]

453

453 = 3 × 151. It is a Blum integer.

454

454 = 2 × 227. It is a nontotient and a Smith number.^[44]

455

455 = 5 × 7 × 13. It is a sphenic number and a tetrahedral number.^[56]

The sum of the squares of the first 455 primes is divisible by 455.^[57]

456

456 = 2³ × 3 × 19. It is a centered pentagonal number,^[58] an icosahedral number, the sum of a pair of twin primes (227 + 229), and the sum of four consecutive primes (107 + 109 + 113 + 127).

It is one of the two values of the magical formula, IPSOS, the other being 696.^[59]

457

457 is a prime number, a self number,^[19] and the sum of three consecutive primes (149 + 151 + 157).

458

458 = 2 × 229. It is a nontotient. There are 458partitions of 24 into divisors of 24.^[60]

459

459 = 3³ × 17. It is a triangular matchstick number.^[61]

460

460 = 2² × 5 × 23. It is a centered triangular number,^[17] a dodecagonal number,^[62] a Harshad number, and the sum of twelve consecutive primes (17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61).

461

461 is a prime number, a Chen prime,^[1] a sexy prime with 467, an Eisenstein prime with no imaginary part, and a prime index prime.

462

462 = 2 × 3 × 7 × 11. Itis a pronic number,^[63] a sparsely totient number,^[64] an idoneal number, a stirling number of the second kind ${\displaystyle \left\{{9 \atop 7}\right\}}$ , the binomial coefficient ${\displaystyle {\tbinom {11}{5}}}$, and the sum of six consecutive primes (67 + 71 + 73 + 79 + 83 + 89).

463

463 is a prime number, a centered heptagonal number,^[65] sum of seven consecutive primes (53 + 59 + 61 + 67 + 71 + 73 + 79). It is the first of seven consecutive primes that are one less than a multiple of 4 (from 463 to 503).

It is a common baseball double play (see baseball positions).

There are 463 days in the synodic period of Ceres.

464

464 = 2⁴ × 29. It is a primitive abundant number.^[66] and the maximal number of pieces that can be obtained by cutting an annulus with 29 cuts.^[40] Since 464 = 21² + 21 + 2, it is the maximum number of regions into which 22 intersecting circles divide the plane.^[31]

465

465 = 3 × 5 × 31. It is a sphenic number, a member of the Padovan sequence,^[67] a Harshad number, and the 30th triangular number.^[68]

466

466 = 2 × 233. It is a noncototient and a lazy caterer number.^[12]

467

467 is a prime number, a safe prime,^[69] a sexy prime with 461, a Chen prime,^[1] and an Eisenstein prime with no imaginary part.

${\displaystyle \sum _{n=0}^{10}{467}^{n}}$ is prime^[21]

468

468 = 2² × 3² × 13. It is a refactorable number,^[33] a self number,^[19] a Harshad number and the sum of ten consecutive primes (29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67).

469

469 = 7 × 67. It is a centered hexagonal number.^[70]

469! - 1 is prime.

470

470 = 2 × 5 × 47. It is a sphenic number, a nontotient, a noncototient, and a cake number.

471

471 = 3 × 157.It is the sum of three consecutive primes (151 + 157 + 163).

It is a perfect totient number,^[71] additionally, φ(471) = φ(σ(471)).^[72]

472

472 = 2³ × 59. It is a nontotient, an untouchable number,^[16] and a refactorable number.^[33] There are 472 distinct ways to cut a 5 × 5 square into squares with integer sides.^[73]

473

473 = 11 × 43. It is a Blum integer and the sum of five consecutive primes (83 + 89 + 97 + 101 + 103).

474

474 = 2 × 3 × 79. It is a sphenic number, a nontotient, a noncototient, an untouchable number,^[16] a nonagonal number,^[74] the sum of the totient function for first 39 integers, and the sum of eight consecutive primes (43 + 47 + 53 + 59 + 61 + 67 + 71 + 73).

475

475 = 5² × 19. It is a 49-gonal number and a member of the Mian–Chowla sequence.^[4]

476

476 = 2² × 7 × 17. It is a Harshad number and an admirable number.^[75]

477

477 = 3² × 53. It is a pentagonal number.^[34]

478

478 = 2 × 239. It is a Companion Pell number. There are 478 partitions of 26 that do not contain 1 as a part.^[76]

479

479 is a prime number, a safe prime,^[69] a Chen prime,^[1] an Eisenstein prime with no imaginary part, a self number,^[19] and the sum of nine consecutive primes (37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71).

480

480 = 2⁵ × 3 × 5. It is a highly totient number,^[37] a refactorable number,^[33] a Harshad number, a largely composite number,^[77] the sum of a twin prime pair (239 + 241), and the sum of four consecutive primes (109 + 113 + 127 + 131).

${\displaystyle \sum _{n=0}^{10}{480}^{n}}$ is prime^[21]

481

481 = 13 × 37. It is an octagonal number,^[15] a centered square number,^[30] and a Harshad number.

482

482 = 2 × 241. It is a nontotient and a noncototient. There are 482 series-reduced planted trees with 15 nodes.^[78]

483

483 = 3 × 7 × 23. It is a sphenic number and a Smith number.^[44]

484

484 = 2² × 11² = 22². It is a palindromic square and a nontotient.

485

485 = 5 × 97. There are 485 triangles (of all sizes, including holes) in Sierpiński's triangle after 5 inscriptions^[79]

486

486 = 2 × 3⁵. It is a Harshad number and a Perrin number.^[80]

487

It is a prime number, a Chen prime,^[1] and the sum of three consecutive primes (157 + 163 + 167).

The only primes under 7.74 × 10¹³ that divide their own decimal repetends are 3, 487, and 56598313.^[81]

488

488 = 2³ × 61. It is a nontotient and a refactorable number.^[33] There are 488 surface points on a cube with edge-length 10.^[82]

φ(488) = φ(σ(488)),^[72]

489

489 = 3 × 163. It is an octahedral number.^[83]

490

490 = 2 × 5 × 7². It is a noncototient, a self number,^[19] and the sum of the totient function for the first 40 integers. There are 490 integer partitions of 19.^[84]

A (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 .

491

491 is a prime number, an isolated prime, a Sophie Germain prime,^[28] a Chen prime,^[1] an Eisenstein prime with no imaginary part, and a strictly non-palindromic number.^[45]

492

492 = 2² × 3 × 41. It is a refactorable number^[33] and the sum of six consecutive primes (71 + 73 + 79 + 83 + 89 + 97). It forms a Ruth–Aaron pair with 493 under first definition.

493

493 = 17 × 29. It is the sum of seven consecutive primes (59 + 61 + 67 + 71 + 73 + 79 + 83). It forms a Ruth–Aaron pair with 492 under first definition. The 493d centered octagonal number is also a centered square number.^[85]

494

494 = 2 × 13 × 19 = ${\displaystyle \left\langle \!\!\left\langle {8 \atop 1}\right\rangle \!\!\right\rangle }$.^[86] It is a sphenic number and a nontotient.

495

496

497

497 = 7 × 71. It is a lazy caterer number^[12] and the sum of five consecutive primes (89 + 97 + 101 + 103 + 107).

498

498 = 2 × 3 × 83. It is a sphenic number, an untouchable number,^[16] an admirable number,^[87] and an abundant number

499

499 is a prime number, an isolated prime, and a Chen prime.^[1]

4⁴⁹⁹ - 3⁴⁹⁹ is prime.