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111 (number)

← 110 111 112 →
Cardinalone hundred eleven
Ordinal111th
(one hundred eleventh)
Factorization3 × 37
Divisors1, 3, 37, 111
Greek numeralΡΙΑ´
Roman numeralCXI
Binary11011112
Ternary110103
Quaternary12334
Quinary4215
Senary3036
Octal1578
Duodecimal9312
Hexadecimal6F16
Vigesimal5B20
Base 363336

111 (One hundred [and] eleven) is the natural number following 110 and preceding 112.

In mathematics

111 is a perfect totient number.[1]

111 is R3 or the second repunit, a number like 11, 111, or 1111 that consists of repeated units, or 1's. It equals 3 × 37, therefore all triplets (numbers like 222 or 777) in base ten are of the form 3n × 37. As a repunit, it also follows that 111 is a palindromic number.

All triplets in all bases are multiples of 111 in that base, therefore the number represented by 111 in a particular base is the only triplet that can ever be prime. 111 is not prime in base ten, but is prime in base two, where 1112 = 710. It is also prime in these other bases up to 128: 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119 (sequence A002384 in the OEIS)

In base 18, the number 111 is 73 (= 34310) which is the only base where 111 is a perfect power.

The smallest magic square using only 1 and prime numbers has a magic constant of 111:

31 73 7
13 37 61
67 1 43

A six-by-six magic square using the numbers 1 to 36 also has a magic constant of 111:

1 11 31 29 19 20
2 22 24 25 8 30
3 33 26 23 17 9
34 27 10 12 21 7
35 14 15 16 18 13
36 4 5 6 28 32

(The square has this magic constant because 1 + 2 + 3 + ... + 34 + 35 + 36 = 666, and 666 / 6 = 111).

111 is also the magic constant of the n-Queens Problem for n = 6.[2] It is also a nonagonal number.[3]

In base 10, it is a Harshad number.[4]

Nelson

In cricket, the number 111 is sometimes called "a Nelson" after Admiral Nelson, who allegedly only had "One Eye, One Arm, One Leg" near the end of his life. This is in fact inaccurate—Nelson never lost a leg. Alternate meanings include "One Eye, One Arm, One Ambition" and "One Eye, One Arm, One Arsehole".

Particularly in cricket, multiples of 111 are called a double Nelson (222), triple Nelson (333), and so on.

A score of 111 is considered by some to be unlucky. To combat the supposed bad luck, some watching lift their feet off the ground. Since an umpire cannot sit down and raise his feet, the international umpire David Shepherd had a whole retinue of peculiar mannerisms if the score was ever a Nelson multiple. He would hop, shuffle, or jiggle, particularly if the number of wickets also matched—111/1, 222/2 etc.

In other fields

111 is also:

See also

References

  1. ^ Sloane, N. J. A. (ed.). "Sequence A082897 (Perfect totient numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-26.
  2. ^ Sloane, N. J. A. (ed.). "Sequence A006003 (a(n) = n*(n^2 + 1)/2)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  3. ^ Sloane, N. J. A. (ed.). "Sequence A001106 (9-gonal (or enneagonal or nonagonal) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-26.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A005349 (Niven (or Harshad) numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-05-26.
  5. ^ John Ronald Reuel Tolkien (1993). The fellowship of the ring: being the first part of The lord of the rings. HarperCollins. ISBN 978-0-261-10235-4.

External links