Wikipedia

1234 (number)

This article is about the number 1234. For the year, see 1234.

1234 is the natural number following 1233, and preceding 1235.

← 1233 1234 1235 →
Cardinalone thousand two hundred thirty-four
Ordinal1234th
(one thousand two hundred thirty-fourth)
Factorization2 × 617
Greek numeral,ΑΣΛΔ´
Roman numeralMCCXXXIV
Binary100110100102
Ternary12002013
Senary54146
Octal23228
Duodecimal86A12
Hexadecimal4D216

A 2012 study of frequently-used personal identification numbers found that, among 4-digit pin codes, 1234 is the most frequently chosen.[1]

Mathematical properties

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Prime factorization

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1234 is a discrete semiprime with distinct prime factors (2, and 617).[2][a]

Concatenation of decimal digits

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1234 is the smallest whole number that contains the digits 1 through 4 in decimal.[b]

The sum of the base-ten digits of 1234 forms the fourth triangular number (10). 1234 is more specifically the fourth member of the "Triangle of the gods" sequence, obtained by concatenating decimal representations of positive integers.[6][7] It is also the fifth member of a related integer sequence, obtained from the recurrence relation   starting from   and  ;[8] both this sequence and the aforementioned sequence begin in the same way, yet they diverge around their tenth positions.

Because it is not divisible by 4, 1234 is the first number in these sequences that is not divisible by its final digit.[9]

Integer partitions

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The number of strict partitions of 44 containing the sum of some subset of the parts is 1234 (as a variation of, sum-full strict partitions).[10]

1234 is also the number of solutions to other integer partitions, including:

  • The number of integer partitions of 24 without all distinct multiplicities.[11]
  • The number of partitions of 332 = 1089 into exactly four prime numbers.[12]
  • The number of partitions of 24 into parts having at most two prime factors.[13]
  • The number of "colored" integer partitions of 12 such that four colors are used and parts differ by size, or by color.[14]

1234 is furthermore the number of partitions of 44 into parts with an odd number of prime divisors (counted with multiplicity).[15]

Binary strings

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1234 is the number of "straight" binary strings of length 22 (i.e., the simplest way of representing quantities with binary numbers), equivalently the number of finite Sturmian words of length 22.[16]

Vertex sets

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There are exactly 1234 independent vertex sets in a 4 × 4 grid.[17] This is equivalent with the ways of choosing a subset of positions in a 4 × 4 grid so that no two chosen positions are adjacent horizontally or vertically.[c]

Notes

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  1. ^ It is the 363rd indexed semiprime,[2] or the 352nd semiprime that is discrete.[3]
  2. ^ The composite index of 1234 is 1031,[4] also the index of the fifth prime repunit in base-ten,   (following  ,  ,  , and  ).[5]
  3. ^ For the corresponding problem in one dimension instead of two (choosing points from a sequence with no two adjacent), the number of solutions represents a Fibonacci number.[17]

References

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  1. ^ Berry, N. (September 3, 2012). "PIN analysis". Data Genetics. As cited by Nisbet, Alastair; Kim, Maria (December 2016). "An analysis of chosen alarm code pin numbers & their weakness against a modified brute force attack". In Johnstone, M. (ed.). The Proceedings of 14th Australian Information Security Management Conference. Perth, Australia: Edith Cowan University. pp. 21–29. doi:10.4225/75/58a69fd2a8b03.{{cite conference}}: CS1 maint: date and year (link)
  2. ^ a b Sloane, N. J. A. (ed.). "Sequence A001358". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  3. ^ Sloane, N. J. A. (ed.). "Sequence A006881". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  4. ^ Sloane, N. J. A. (ed.). "Sequence A002808". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  5. ^ Sloane, N. J. A. (ed.). "Sequence A004023". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  6. ^ Sloane, N. J. A. (ed.). "Sequence A007908 (Triangle of the gods)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  7. ^ Pickover, Clifford A. (2011). A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality. Turner Publishing Company. pp. 10–11. ISBN 9781118046074.
  8. ^ Sloane, N. J. A. (ed.). "Sequence A014824". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  9. ^ Parker, Matt (2014). Things to Make and Do in the Fourth Dimension: A Mathematician's Journey Through Narcissistic Numbers, Optimal Dating Algorithms, at Least Two Kinds of Infinity, and More. Macmillan. p. 8. ISBN 9780374275655.
  10. ^ Sloane, N. J. A. (ed.). "Sequence A364272". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  11. ^ Sloane, N. J. A. (ed.). "Sequence A336866". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  12. ^ Sloane, N. J. A. (ed.). "Sequence A243940". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  13. ^ Sloane, N. J. A. (ed.). "Sequence A101049". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  14. ^ Sloane, N. J. A. (ed.). "Sequence A327382". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  15. ^ Sloane, N. J. A. (ed.). "Sequence A286218". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  16. ^ Sloane, N. J. A. (ed.). "Sequence A005598". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
  17. ^ a b Sloane, N. J. A. (ed.). "Sequence A006506". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
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