Strobogrammatic number

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A strobogrammatic number is a number that appears the same when rotated 180 degrees, that is, it reads the same right-side up and upside down (e.g., 69, 96, 1001).[1] A strobogrammatic prime is a strobogrammatic number that is also a prime number, i.e., a number that is only divisible by one and itself (e.g., 11).[2] It is a type of ambigram, words and numbers that retain their meaning when viewed from a different perspective, such as palindromes.[3]

When written using standard characters (ASCII), the numbers, 0, 1, 8 are symmetrical around the horizontal axis, and 6 and 9 are the same as each other when rotated 180 degrees. In such a system, the first few strobogrammatic numbers are:

0, 1, 8, 11, 69, 88, 96, 101, 111, 181, 609, 619, 689, 808, 818, 888, 906, 916, 986, 1001, 1111, 1691, 1881, 1961, 6009, 6119, 6699, 6889, 6969, 8008, 8118, 8698, 8888, 8968, 9006, 9116, 9696, 9886, 9966, ... (sequence A000787 in the OEIS)

The first few strobogrammatic primes are:

11, 101, 181, 619, 16091, 18181, 19861, 61819, 116911, 119611, 160091, 169691, 191161, 196961, 686989, 688889, ... (sequence A007597 in the OEIS)

The years 1881 and 1961 were the most recent strobogrammatic years; the next strobogrammatic year will be 6009.

Although amateur aficionados of mathematics are quite interested in this concept, professional mathematicians generally are not. Like the concept of repunits and palindromic numbers, the concept of strobogrammatic numbers is base-dependent (expanding to base-sixteen, for example, produces the additional symmetries of 3/E; some variants of duodecimal systems also have this and a symmetrical x). Unlike palindromes, it is also font dependent. But the concept of strobogrammatic numbers is not neatly expressible algebraically, the way that the concept of repunits is, or even the concept of palindromic numbers.

Nonstandard systems[edit]

The strobogrammatic properties of a given number vary by font. For instance, in an ornate serif font, the numbers 2 and 7 may be rotations of each other; however, in a seven-segment display emulator, this correspondence is lost, but 2 and 5 are both symmetrical. There are sets of glyphs for writing numbers in base 10, such as the Devanagari and Gurmukhi of India in which the numbers listed above are not strobogrammatic at all.

In binary, given a glyph for 1 consisting of a single line without hooks or serifs, all palindromic numbers are strobogrammatic (as well as dihedral), which means (among other things) that all Mersenne numbers are strobogrammatic. Dihedral primes that do not use 2 or 5 are also strobogrammatic primes.

In duodecimal, they are (using inverted two and three for ten and eleven, respectively)

0, 1, 8, 11, 2ᘔ, 3Ɛ, 69, 88, 96, ᘔ2, Ɛ3, 101, 111, 181, 20ᘔ, 21ᘔ, 28ᘔ, 30Ɛ, 31Ɛ, 38Ɛ, 609, 619, 689, 808, 818, 888, 906, 916, 986, ᘔ02, ᘔ12, ᘔ82, Ɛ03, Ɛ13, Ɛ83, ...

Examples of strobogrammatic primes in duodecimal are:

11, 3Ɛ, 111, 181, 30Ɛ, 12ᘔ1, 13Ɛ1, 311Ɛ, 396Ɛ, 3ᘔ2Ɛ, 11111, 11811, 130Ɛ1, 16191, 18881, 1Ɛ831, 3000Ɛ, 3181Ɛ, 328ᘔƐ, 331ƐƐ, 338ƐƐ, 3689Ɛ, 3818Ɛ, 3888Ɛ, ...

References[edit]

  1. ^ Schaaf, William L. (March 1, 2016) [1999]. "Number game". Encyclopedia Britannica. Retrieved 22 January 2017. 
  2. ^ Caldwell, Chris K. "The Prime Glossary: strobogrammatic". primes.utm.edu. Retrieved 22 January 2017. 
  3. ^ "A000787 - OEIS". The On-Line Encyclopedia of Integer Sequences. Retrieved 22 January 2017. 

External links[edit]