500 (number)
This articleneeds additional citations forverification.(January 2019) |
| ||||
---|---|---|---|---|
Cardinal | five hundred | |||
Ordinal | 500th (five hundredth) | |||
Factorization | 22× 53 | |||
Greek numeral | Φ´ | |||
Roman numeral | D | |||
Binary | 1111101002 | |||
Ternary | 2001123 | |||
Senary | 21526 | |||
Octal | 7648 | |||
Duodecimal | 35812 | |||
Hexadecimal | 1F416 | |||
Armenian | Շ | |||
Hebrew | ת "ק / ך | |||
Babylonian cuneiform | 𒐜⟪ | |||
Egyptian hieroglyph | 𓍦 |
500(five hundred) is thenatural numberfollowing499and preceding501.
Mathematical properties
[edit]500 = 22× 53.It is anAchilles numberand aHarshad number,meaning it is divisible by the sum of its digits. It is the number of planar partitions of 10.[1]
Other fields
[edit]Five hundredis also
- the number that manyNASCARraces often use at the end of their race names (e.g.,Daytona 500), to denote the length of the race (inmiles,kilometersor laps).
- the longest advertised distance (in miles) of theIndyCar Seriesand its premier race, theIndianapolis 500.
Slang names
[edit]- Monkey (UK slang for £500; US slang for $500)[2]
Integers from 501 to 599
[edit]500s
[edit]501
[edit]501 = 3 × 167. It is:
- the sum of the first 18 primes (a term of the sequenceOEIS:A007504).
- palindromic in bases 9 (6169) and 20 (15120).
502
[edit]503
[edit]503 is:
- aprime number.
- asafe prime.[3]
- the sum of three consecutive primes (163 + 167 + 173).[4]
- the sum of the cubes of the first four primes.[5]
- aChen prime[6]
- anEisenstein primewith no imaginary part.[7]
- an index of a prime Lucas number.[8]
- an isolated prime
504
[edit]504 = 23× 32× 7. It is:
- the sum between the smallest pair ofamicable numbers(220, 284).[9]
- atribonacci number.[10]
- asemi-meandric number.
- a refactorable number.[11]
- a Harshad number.
- is prime[12]
- the group order of the fourth smallestnon-cyclicsimple groupA1(8) =2G2(3)′.
- the number of symmetries of thesimple groupPSL(2,8)that is theautomorphism groupof theMacbeath surface.[13]
- alargely composite number[14]
505
[edit]- 505 = 5 × 101
- model number ofLevi'sjeans,model number ofU-505
- This number is themagic constantofn×nnormalmagic squareandn-queens problemforn= 10.
506
[edit]506 = 2 × 11 × 23. It is:
- asphenic number.
- asquare pyramidal number.[15]
- apronic number.[16]
- a Harshad number.
is a prime number. Its decimal expansion is 252 nines, an eight, and 253 more nines.
507
[edit]- 507 = 3 × 132= 232- 23 + 1, which makes it a central polygonal number[17]
- The ageMinghad before dying.
508
[edit]- 508 = 22× 127, sum of four consecutive primes (113 + 127 + 131 + 137), number of graphical forest partitions of 30,[18]since 508 = 222+ 22 + 2 it is the maximum number of regions into which 23intersecting circles divide the plane.[19]
509
[edit]509 is:
- a prime number.
- aSophie Germain prime,smallest Sophie Germain prime to start a 4-termCunningham chainof the first kind {509, 1019, 2039, 4079}.
- a Chen prime.
- an Eisenstein prime with no imaginary part.
- ahighly cototient number[20]
- aprime index prime.
510s
[edit]510
[edit]510 = 2 × 3 × 5 × 17. It is:
- the sum of eight consecutive primes (47 + 53 + 59 + 61 + 67 + 71 + 73 + 79).
- the sum of ten consecutive primes (31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71).
- the sum of twelve consecutive primes (19 + 23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67).
- anontotient.
- asparsely totient number.[21]
- a Harshad number.
- the number of nonempty proper subsets of an 9-element set.[22]
511
[edit]511 = 7 × 73. It is:
- a Harshad number.
- a palindromic number and arepdigitin bases 2 (1111111112) and 8 (7778)
- 5-1-1,a roadway status andtransitinformationhotlinein many metropolitan areas of theUnited States.
512
[edit]512 = 83= 29.It is:
- apower of two
- acubeof8
- aLeyland number[23]using 4 & 4 (44+ 44)
- aDudeney number.[24]
- a Harshad number
- palindromic in bases 7 (13317) and 15 (24215)
- a vertically symmetric number (sequenceA053701in theOEIS)
513
[edit]513 = 33× 19. It is:
- Leyland number of the second kind[25]using 3 & 6 (36- 63)
- palindromic in bases 2 (10000000012) and 8 (10018)
- a Harshad number
- Area code of Cincinnati, Ohio
514
[edit]514 = 2 × 257, it is:
- acentered triangular number.[26]
- anontotient
- a palindrome in bases 4 (200024), 16 (20216), and 19 (18119)
- an Area Code for Montreal, Canada
515
[edit]515 = 5 × 103, it is:
- the sum of nine consecutive primes (41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73).
- the number of complete compositions of 11.[27]
516
[edit]516 = 22× 3 × 43, it is:
- nontotient.
- untouchable number.[28]
- refactorable number.[11]
- a Harshad number.
517
[edit]517 = 11 × 47, it is:
- the sum of five consecutive primes (97 + 101 + 103 + 107 + 109).
- aSmith number.[29]
518
[edit]518 = 2 × 7 × 37, it is:
- = 51+ 12+ 83(a property shared with175and 598).
- a sphenic number.
- a nontotient.
- an untouchable number.[28]
- palindromic and a repdigit in bases 6 (22226) and 36 (EE36).
- a Harshad number.
519
[edit]519 = 3 × 173, it is:
- the sum of three consecutive primes (167 + 173 + 179)
- palindromic in bases 9 (6369) and 12 (37312)
- aD-number.[30]
520s
[edit]520
[edit]520 = 23× 5 × 13. It is:
- anuntouchable number.[28]
- anidoneal number
- a palindromic number in base 14 (29214).
521
[edit]521 is:
- aLucas prime.[31]
- AMersenne exponent,i.e. 2521−1 is prime.
- The largest known such exponent that is the lesser oftwin primes[32]
- a Chen prime.
- an Eisenstein prime with no imaginary part.
- palindromic in bases 11 (43411) and 20 (16120).
522
[edit]522 = 2 × 32× 29. It is:
- the sum of six consecutive primes (73 + 79 + 83 + 89 + 97 + 101).
- a repdigit in bases 28 (II28) and 57 (9957).
- a Harshad number.
- number of series-parallel networks with 8 unlabeled edges.[33]
523
[edit]523 is:
- a prime number.
- the sum of seven consecutive primes (61 + 67 + 71 + 73 + 79 + 83 + 89).
- palindromic in bases 13 (31313) and 18 (1B118).
- aprime with a prime number of prime digits[34]
- the smallest prime number that starts aprime gapof length greater than 14
524
[edit]524 = 22× 131
- number of partitions of 44 into powers of 2[35]
525
[edit]525 = 3 × 52× 7. It ispalindromicin base ten, as well as the fifty-fifthself numbergreater than 1 indecimal.[36]It is also:
- the sum of all prime numbers that divide theordersof the twenty-sixsporadic groups(2, 3, 5,..., 71; aside from 53 and 61).[37]
- the sum of the dimensions of all fiveexceptional Lie algebras(14, 52, 78, 133, 248).[38]
525 is the number of scan lines in theNTSCtelevision standard.
526
[edit]526 = 2 × 263,centered pentagonal number,[39]nontotient, Smith number[29]
527
[edit]527 = 17 × 31. It is:
- palindromic in base 15 (25215)
- number of diagonals in a 34-gon[40]
- also, the section of the US Tax Code regulatingsoft moneypolitical campaigning (see527 groups)
528
[edit]528 = 24× 3 × 11. It is:
- atriangular number.
- palindromic in bases 9 (6469) and 17 (1E117).
529
[edit]529 = 232.It is:
- acentered octagonal number.[41]
- a lazy caterer number (sequenceA000124in theOEIS).
- also Section 529 of the IRS tax code organizes529 plansto encourage saving for higher education.
530s
[edit]530
[edit]530 = 2 × 5 × 53. It is:
- asphenic number.
- anontotient.
- the sum of totient function for first 41integers.
- anuntouchable number.[28]
- the sum of the first threeperfect numbers.
- palindromic in bases 4 (201024), 16 (21216), and 23 (10123).
- a US telephonearea codethat covers much ofNorthern California.
531
[edit]531 = 32× 59. It is:
- palindromic in base 12 (38312).
- a Harshad number.
- number of symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to 6[42]
532
[edit]532 = 22× 7 × 19. It is:
- apentagonal number.[43]
- a nontotient.
- palindromic and a repdigit in bases 11 (44411), 27 (JJ27), and 37 (EE37).
- admirable number.
533
[edit]533 = 13 × 41. It is:
- the sum of three consecutive primes (173 + 179 + 181).
- the sum of five consecutive primes (101 + 103 + 107 + 109 + 113).
- palindromic in base 19 (19119).
- generalized octagonal number.[44]
534
[edit]534 = 2 × 3 × 89. It is:
- a sphenic number.
- the sum of four consecutive primes (127 + 131 + 137 + 139).
- a nontotient.
- palindromic in bases 5 (41145) and 14 (2A214).
- anadmirable number.
- is prime[12]
535
[edit]535 = 5 × 107. It is:
- a Smith number.[29]
for;this polynomial plays an essential role inApéry's proofthatis irrational.
535 is used as an abbreviation for May 35, which is used in China instead of June 4 to evade censorship by the Chinese government of references on the Internet to theTiananmen Square protests of 1989.[45]
536
[edit]536 = 23× 67. It is:
- the number of ways to arrange the pieces of theostomachioninto a square, not counting rotation or reflection.
- the number of 1's in all partitions of 23 into odd parts[46]
- a refactorable number.[11]
- the lowesthappy numberbeginning with the digit 5.
537
[edit]537 = 3 × 179,Mertens function(537) = 0,Blum integer,D-number[30]
538
[edit]538 = 2 × 269. It is:
- anopen meandric number.
- a nontotient.
- the total number of votes in theUnited States Electoral College.
- the websiteFiveThirtyEight.
- Radio 538,a Dutch commercial radio station
539
[edit]539 = 72× 11
is prime[12]
540s
[edit]540
[edit]540 = 22× 33× 5. It is:
- anuntouchable number.[28]
- aheptagonal number.
- adecagonal number.[47]
- a repdigit in bases 26 (KK26), 29 (II29), 35 (FF35), 44 (CC44), 53 (AA53), and 59 (9959).
- a Harshad number.
- the number of doors toValhallaaccording to theProse Edda.[48]
- the number of floors inThor'shall, known asBilskirnir,according to theProse Edda.[49]
- the sum of a twin prime (269 + 271)
- alargely composite number[14]
541
[edit]541 is:
- the 100th prime.
- alucky prime.[50]
- a Chen prime.
- the 10thstar number.[51]
- palindromic in bases 18 (1C118) and 20 (17120).
- the fifthordered Bell numberthat represents the number ofordered partitionsof.[52]
- 4541- 3541is prime.[53]
For theMertens function,
542
[edit]542 = 2 × 271. It is:
- anontotient.
- the sum oftotient functionfor the first 42 integers.[54]
543
[edit]543 = 3 × 181; palindromic in bases 11 (45411) and 12 (39312),D-number.[30]
is prime[12]
544
[edit]544 = 25× 17. Take a grid of 2 times 5 points. There are 14 points on the perimeter. Join every pair of the perimeter points by a line segment. The lines do not extend outside the grid.544 is the number of regions formed by these lines.OEIS:A331452
544 is also the number of pieces that could be seen in a5×5×5×5 Rubik's Tesseract.As a standard 5×5×5 has 98 visible pieces (53− 33), a 5×5×5×5 has 544 visible pieces (54− 34).
545
[edit]545 = 5 × 109. It is:
- acentered square number.[55]
- palindromic in bases 10 (54510) and 17 (1F117).
546
[edit]546 = 2 × 3 × 7 × 13. It is:
- the sum of eight consecutive primes (53 + 59 + 61 + 67 + 71 + 73 + 79 + 83).
- palindromic in bases 4 (202024), 9 (6669), and 16 (22216).
- a repdigit in bases 9 and 16.
- 546! − 1 is prime.
547
[edit]547 is:
- a prime number.
- acuban prime.[56]
- acentered hexagonal number.[57]
- acentered heptagonal number.[58]
- aprime index prime.
548
[edit]548 = 22× 137. It is:
- anontotient.
- the default port for theApple Filing Protocol.
Also, every positive integer is the sum of at most 548 ninth powers;
549
[edit]549 = 32× 61, it is:
- a repdigit in bases 13 (33313) and 60 (9960).
- φ(549) = φ(σ(549)).[59]
550s
[edit]550
[edit]550 = 2 × 52× 11. It is:
- apentagonal pyramidal number.[60]
- aprimitive abundant number.[61]
- a nontotient.
- a repdigit in bases 24 (MM24), 49 (BB49), and 54 (AA54).
- a Harshad number.
- the SMTP status code meaning the requested action was not taken because the mailbox is unavailable
551
[edit]551 = 19 × 29. It is:
- It is the number of mathematicaltreeson 12 unlabeled nodes.[62]
- the sum of three consecutive primes (179 + 181 + 191).
- palindromic in base 22 (13122).
- theSMTPstatus code meaning user is not local
552
[edit]552 = 23× 3 × 23. It is:
- the number of prime knots with 11 crossings.[63]
- the sum of six consecutive primes (79 + 83 + 89 + 97 + 101 + 103).
- the sum of ten consecutive primes (37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73).
- a pronic number.[16]
- an untouchable number.[28]
- palindromic in base 19 (1A119).
- a Harshad number.
- the model number ofU-552.
- the SMTP status code meaning requested action aborted because the mailbox is full.
553
[edit]553 = 7 × 79. It is:
- the sum of nine consecutive primes (43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79).
- central polygonal number.[17]
- the model number ofU-553.
- theSMTP status codemeaning requested action aborted because of faulty mailbox name.
554
[edit]554 = 2 × 277. It is:
- a nontotient.
- a2-Knödel number
- the SMTP status code meaning transaction failed.
Mertens function(554) = 6, a record high that stands until 586.
555
[edit]555 = 3 × 5 × 37 is:
- asphenic number.
- palindromic in bases 9 (6769), 10 (55510), and 12 (3A312).
- a repdigit in bases 10 and 36.
- a Harshad number.
- φ(555) = φ(σ(555)).[59]
556
[edit]556 = 22× 139. It is:
- the sum of four consecutive primes (131 + 137 + 139 + 149).
- anuntouchable number,because it is never the sum of the proper divisors of any integer.[28]
- a happy number.
- the model number ofU-556;5.56×45mm NATOcartridge.
557
[edit]557 is:
- a prime number.
- a Chen prime.
- an Eisenstein prime with no imaginary part.
- the number of parallelogram polyominoes with 9 cells.[64]
558
[edit]558 = 2 × 32× 31. It is:
- a nontotient.
- a repdigit in bases 30 (II30) and 61 (9961).
- a Harshad number.
- The sum of the largest prime factors of the first 558 is itself divisible by 558 (the previous such number is 62, the next is 993).
- in the title of theStar Trek: Deep Space Nineepisode "The Siege of AR-558"
559
[edit]559 = 13 × 43. It is:
- the sum of five consecutive primes (103 + 107 + 109 + 113 + 127).
- the sum of seven consecutive primes (67 + 71 + 73 + 79 + 83 + 89 + 97).
- anonagonal number.[65]
- acentered cube number.[66]
- palindromic in base 18 (1D118).
- the model number ofU-559.
560s
[edit]560
[edit]560 = 24× 5 × 7. It is:
- atetrahedral number.[67]
- a refactorable number.
- palindromic in bases 3 (2022023) and 6 (23326).
- the number of diagonals in a 35-gon[40]
561
[edit]561 = 3 × 11 × 17. It is:
- asphenic number.
- a triangular number.
- ahexagonal number.[68]
- palindromic in bases 2 (10001100012) and 20 (18120).
- the firstCarmichael number[69]
562
[edit]562 = 2 × 281. It is:
- a Smith number.[29]
- an untouchable number.[28]
- the sum of twelve consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47 + 53 + 59 + 61 + 67 + 71).
- palindromic in bases 4 (203024), 13 (34313), 14 (2C214), 16 (23216), and 17 (1G117).
- a lazy caterer number (sequenceA000124in theOEIS).
- the number of Native American (including Alaskan) Nations, or "Tribes," recognized by the USA government.
563
[edit]563 is:
- a prime number.
- asafe prime.[3]
- the largest knownWilson prime.[70]
- a Chen prime.
- an Eisenstein prime with no imaginary part.
- abalanced prime.[71]
- a strictly non-palindromic number.[72]
- asexy prime.
- a happy prime.
- aprime index prime.
- 5563- 4563is prime.[73]
564
[edit]564 = 22× 3 × 47. It is:
- the sum of a twin prime (281 + 283).
- a refactorable number.
- palindromic in bases 5 (42245) and 9 (6869).
- number of primes <= 212.[74]
565
[edit]565 = 5 × 113. It is:
- the sum of three consecutive primes (181 + 191 + 193).
- a member of theMian–Chowla sequence.[75]
- a happy number.
- palindromic in bases 10 (56510) and 11 (47411).
566
[edit]566 = 2 × 283. It is:
- nontotient.
- a happy number.
- a2-Knödel number.
567
[edit]567 = 34× 7. It is:
- palindromic in base 12 (3B312).
- is prime[12]
568
[edit]568 = 23× 71. It is:
- the sum of the first nineteen primes (a term of the sequenceOEIS:A007504).
- a refactorable number.
- palindromic in bases 7 (14417) and 21 (16121).
- the smallest number whose seventh power is the sum of 7 seventh powers.
- the room number booked byBenjamin Braddockin the 1967 filmThe Graduate.
- the number of millilitres in animperial pint.
- the name of the Student Union bar atImperial College London
569
[edit]569 is:
- a prime number.
- a Chen prime.
- an Eisenstein prime with no imaginary part.
- a strictly non-palindromic number.[72]
570s
[edit]570
[edit]570 = 2 × 3 × 5 × 19. It is:
571
[edit]571 is:
- a prime number.
- a Chen prime.
- a centered triangular number.[26]
- the model number ofU-571which appeared in the 2000 movieU-571
572
[edit]572 = 22× 11 × 13. It is:
- aprimitive abundant number.[61]
- a nontotient.
- palindromic in bases 3 (2100123) and 15 (28215).
573
[edit]573 = 3 × 191. It is:
- aBlum integer
- known as theKonami number,since "ko-na-mi" is associated with 573 in the Japanese wordplayGoroawase
- the model number ofGerman submarineU-573
574
[edit]574 = 2 × 7 × 41. It is:
- a sphenic number.
- a nontotient.
- palindromic in base 9 (7079).
- number of partitions of 27 that do not contain 1 as a part.[78]
- number of amino acid residues in ahemoglobinmolecule.
575
[edit]575 = 52× 23. It is:
- palindromic in bases 10 (57510) and 13 (35313).
- acentered octahedral number.[79]
And the sum of the squares of the first 575 primes is divisible by 575.[80]
576
[edit]576 = 26× 32= 242.It is:
- the sum of four consecutive primes (137 + 139 + 149 + 151).
- ahighly totient number.[81]
- a Smith number.[29]
- an untouchable number.[28]
- palindromic in bases 11 (48411), 14 (2D214), and 23 (12123).
- a Harshad number.
- four-dozen sets of a dozen, which makes it 4 gross.
- acake number.
- the number of parts in all compositions of 8.[82]
577
[edit]577 is:
- a prime number.
- aProth prime.[83]
- a Chen prime.
- palindromic in bases 18 (1E118) and 24 (10124).
- the number of seats inNational Assembly (France).
578
[edit]578 = 2 × 172.It is:
- a nontotient.
- palindromic in base 16 (24216).
- area of a square with diagonal 34[84]
579
[edit]579 = 3 × 193; it is aménage number,[85]and asemiprime.
580s
[edit]580
[edit]580 = 22× 5 × 29. It is:
- the sum of six consecutive primes (83 + 89 + 97 + 101 + 103 + 107).
- palindromic in bases 12 (40412) and 17 (20217).
581
[edit]581 = 7 × 83. It is:
- the sum of three consecutive primes (191 + 193 + 197).
- aBlum integer
582
[edit]582 = 2 × 3 × 97. It is:
- a sphenic number.
- the sum of eight consecutive primes (59 + 61 + 67 + 71 + 73 + 79 + 83 + 89).
- a nontotient.
- a vertically symmetric number (sequenceA053701in theOEIS).
- anadmirable number.
583
[edit]583 = 11 × 53. It is:
- palindromic in base 9 (7179).
- number of compositions of 11 whose run-lengths are either weakly increasing or weakly decreasing[86]
584
[edit]584 = 23× 73. It is:
- an untouchable number.[28]
- the sum of totient function for first 43 integers.
- a refactorable number.
585
[edit]585 = 32× 5 × 13. It is:
- palindromic in bases 2 (10010010012), 8 (11118), and 10 (58510).
- a repdigit in bases 8, 38, 44, and 64.
- the sum of powers of 8 from 0 to 3.
When counting in binary with fingers, expressing 585 as 1001001001, results in the isolation of the index and little fingers of each hand, "throwing up thehorns".
586
[edit]586 = 2 × 293.
- Mertens function(586) = 7 a record high that stands until 1357.
- 2-Knödel number.
- it is the number of several popular personal computer processors (such as the IntelPentium).
587
[edit]587 is:
- a prime number.
- safe prime.[3]
- a Chen prime.
- an Eisenstein prime with no imaginary part.
- the sum of five consecutive primes (107 + 109 + 113 + 127 + 131).
- palindromic in bases 11 (49411) and 15 (29215).
- the outgoing port foremailmessage submission.
- aprime index prime.
588
[edit]588 = 22× 3 × 72.It is:
- a Smith number.[29]
- palindromic in base 13 (36313).
- a Harshad number.
589
[edit]589 = 19 × 31. It is:
- the sum of three consecutive primes (193 + 197 + 199).
- palindromic in base 21 (17121).
- acentered tetrahedral number.
590s
[edit]590
[edit]590 = 2 × 5 × 59. It is:
- a sphenic number.
- apentagonal number.[43]
- a nontotient.
- palindromic in base 19 (1C119).
591
[edit]592
[edit]592 = 24× 37. It is:
- palindromic in bases 9 (7279) and 12 (41412).
- a Harshad number.
593
[edit]593 is:
- a prime number.
- aSophie Germain prime.
- the sum of seven consecutive primes (71 + 73 + 79 + 83 + 89 + 97 + 101).
- the sum of nine consecutive primes (47 + 53 + 59 + 61 + 67 + 71 + 73 + 79 + 83).
- anEisenstein primewith no imaginary part.
- abalanced prime.[71]
- aLeyland prime[87]using 2 & 9 (29+ 92)
- a member of the Mian–Chowla sequence.[75]
- a strictly non-palindromic number.[72]
594
[edit]594 = 2 × 33× 11. It is:
- the sum of ten consecutive primes (41 + 43 + 47 + 53 + 59 + 61 + 67 + 71 + 73 + 79).
- a nontotient.
- palindromic in bases 5 (43345) and 16 (25216).
- a Harshad number.
- the number of diagonals in a 36-gon.[40]
- a balanced number.[77]
595
[edit]595 = 5 × 7 × 17. It is:
- a sphenic number.
- a triangular number.
- centered nonagonal number.[88]
- palindromic in bases 10 (59510) and 18 (1F118).
596
[edit]596 = 22× 149. It is:
- the sum of four consecutive primes (139 + 149 + 151 + 157).
- a nontotient.
- a lazy caterer number (sequenceA000124in theOEIS).
597
[edit]597 = 3 × 199. It is:
598
[edit]598 = 2 × 13 × 23 = 51+ 92+ 83.It is:
- a sphenic number.
- palindromic in bases 4 (211124) and 11 (4A411).
- number of non-alternating permutations of {1...6}.
599
[edit]599 is:
- a prime number.
- a Chen prime.
- an Eisenstein prime with no imaginary part.
- aprime index prime.
References
[edit]- ^Sloane, N. J. A.(ed.)."Sequence A000219 (Number of planar partitions (or plane partitions) of n)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Evans, I.H.,Brewer's Dictionary of Phrase and Fable,14th ed., Cassell, 1990,ISBN0-304-34004-9
- ^abcSloane, N. J. A.(ed.)."Sequence A005385 (Safe primes)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^that is, a term of the sequenceOEIS:A034961
- ^that is, the first term of the sequenceOEIS:A133525
- ^since 503+2 is a product of two primes, 5 and 101
- ^since it is a prime which is congruent to 2 modulo 3.
- ^Sloane, N. J. A.(ed.)."Sequence A001606 (Indices of prime Lucas numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A259180 (Amicable pairs.)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2024-05-22.
- ^Sloane, N. J. A.(ed.)."Sequence A000073 (Tribonacci numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^abcSloane, N. J. A.(ed.)."Sequence A033950 (Refactorable numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^abcdeSloane, N. J. A.(ed.)."Sequence A162862 (Numbers n such that n^10 + n^9 + n^8 + n^7 + n^6 + n^5 + n^4 + n^3 + n^2 + n + 1 is prime)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2022-06-02.
- ^Wohlfahrt, K. (1985)."Macbeath's curve and the modular group".Glasgow Math. J.27:239–247.doi:10.1017/S0017089500006212.MR0819842.
- ^abSloane, N. J. A.(ed.)."Sequence A067128 (Ramanujan's largely composite numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A000330 (Square pyramidal numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^abSloane, N. J. A.(ed.)."Sequence A002378 (Oblong (or promic, pronic, or heteromecic) numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^abSloane, N. J. A.(ed.)."Sequence A002061".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A000070".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2022-05-31.
- ^Sloane, N. J. A.(ed.)."Sequence A014206".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A100827 (Highly cototient numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A036913 (Sparsely totient numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A000918".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A076980 (Leyland numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A061209 (Numbers which are the cubes of their digit sum)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A045575 (Leyland numbers of the second kind)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^abSloane, N. J. A.(ed.)."Sequence A005448 (Centered triangular numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A107429 (Number of complete compositions of n)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^abcdefghijSloane, N. J. A.(ed.)."Sequence A005114 (Untouchable numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^abcdefSloane, N. J. A.(ed.)."Sequence A006753 (Smith numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^abcdSloane, N. J. A.(ed.)."Sequence A033553 (3-Knödel numbers or D-numbers: numbers n > 3 such that n)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2022-05-31.
- ^Sloane, N. J. A.(ed.)."Sequence A005479 (Prime Lucas numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Dr. Kirkby (May 19, 2021)."Many more twin primes below Mersenne exponents than above Mersenne exponents".Mersenne Forum.
- ^Sloane, N. J. A.(ed.)."Sequence A000084 (Number of series-parallel networks with n unlabeled edges. Also called yoke-chains by Cayley and MacMahon.)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A348699 (Primes with a prime number of prime digits)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A000123 (Number of binary partitions: number of partitions of 2n into powers of 2)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A003052 (Self numbers or Colombian numbers (numbers that are not of the form m + sum of digits of m for any m).)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2024-01-09.
- ^Sloane, N. J. A.(ed.)."Sequence A329191 (The prime divisors of the orders of the sporadic finite simple groups.)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2024-01-09.
- ^Sloane, N. J. A.(ed.)."Sequence A113907 (Dimensions of the five sporadic Lie groups.)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2024-01-09.
- ^Sloane, N. J. A.(ed.)."Sequence A005891 (Centered pentagonal numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^abcSloane, N. J. A.(ed.)."Sequence A000096".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2022-05-31.
- ^Sloane, N. J. A.(ed.)."Sequence A016754 (Odd squares: a(n) = (2n+1)^2. Also centered octagonal numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A138178 (Number of symmetric matrices with nonnegative integer entries and without zero rows or columns such that sum of all entries is equal to n)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^abSloane, N. J. A.(ed.)."Sequence A000326 (Pentagonal numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A001082 (Generalized octagonal numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Larmer, Brook (October 26, 2011)."Where an Internet Joke Is Not Just a Joke".New York Times.RetrievedNovember 1,2011.
- ^Sloane, N. J. A.(ed.)."Sequence A036469 (Partial sums of A000009 (partitions into distinct parts))".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A001107 (10-gonal (or decagonal) numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Snorri Sturluson (1880)."Prose Edda".p. 107.
- ^Snorri Sturluson (1880)."Prose Edda".p. 82.
- ^Sloane, N. J. A.(ed.)."Sequence A031157 (Numbers that are both lucky and prime)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A003154 (Centered 12-gonal numbers. Also star numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A000670 (Fubini numbers: number of preferential arrangements of n labeled elements; or number of weak orders on n labeled elements; or number of ordered partitions of [n].)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2023-10-23.
- ^Sloane, N. J. A.(ed.)."Sequence A059801 (Numbers k such that 4^k - 3^k is prime.)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2023-10-23.
- ^Sloane, N. J. A.(ed.)."Sequence A002088".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A001844 (Centered square numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A002407 (Cuban primes)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A003215 (Hex (or centered hexagonal) numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A069099 (Centered heptagonal numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^abSloane, N. J. A.(ed.)."Sequence A006872".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A002411 (Pentagonal pyramidal numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^abSloane, N. J. A.(ed.)."Sequence A071395 (Primitive abundant numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^"Sloane's A000055: Number of trees with n unlabeled nodes".The On-Line Encyclopedia of Integer Sequences.OEIS Foundation.Archivedfrom the original on 2010-11-29.Retrieved2021-12-19.
- ^Sloane, N. J. A.(ed.)."Sequence A002863 (Number of prime knots with n crossings)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A006958 (Number of parallelogram polyominoes with n cells (also called staircase polyominoes, although that term is overused))".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A001106 (9-gonal (or enneagonal or nonagonal) numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A005898 (Centered cube numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A000292 (Tetrahedral numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A000384 (Hexagonal numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Higgins, Peter (2008).Number Story: From Counting to Cryptography.New York: Copernicus. p.14.ISBN978-1-84800-000-1.
- ^Sloane, N. J. A.(ed.)."Sequence A007540 (Wilson primes)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^abSloane, N. J. A.(ed.)."Sequence A006562 (Balanced primes)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^abcSloane, N. J. A.(ed.)."Sequence A016038 (Strictly non-palindromic numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A059802 (Numbers k such that 5^k - 4^k is prime)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A007053".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2022-06-02.
- ^abSloane, N. J. A.(ed.)."Sequence A005282 (Mian-Chowla sequence)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A045943".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2022-06-02.
- ^abSloane, N. J. A.(ed.)."Sequence A020492 (Balanced numbers: numbers k such that phi(k) (A000010) divides sigma(k) (A000203))".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A002865 (Number of partitions of n that do not contain 1 as a part)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2022-06-02.
- ^Sloane, N. J. A.(ed.)."Sequence A001845 (Centered octahedral numbers (crystal ball sequence for cubic lattice))".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2022-06-02.
- ^Sloane, N. J. A.(ed.)."Sequence A111441 (Numbers k such that the sum of the squares of the first k primes is divisible by k)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2022-06-02.
- ^Sloane, N. J. A.(ed.)."Sequence A097942 (Highly totient numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A001792".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A080076 (Proth primes)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A001105".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A000179 (Ménage numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.
- ^Sloane, N. J. A.(ed.)."Sequence A332835 (Number of compositions of n whose run-lengths are either weakly increasing or weakly decreasing)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2022-06-02.
- ^Sloane, N. J. A.(ed.)."Sequence A094133 (Leyland prime numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.
- ^Sloane, N. J. A.(ed.)."Sequence A060544 (Centered 9-gonal (also known as nonagonal or enneagonal) numbers)".TheOn-Line Encyclopedia of Integer Sequences.OEIS Foundation.Retrieved2016-06-11.