142857

Natural number, cyclic number 1/7 (number) 7 (number) February 2026 142857 1, 3, 9, 11, 13, 27, 33, 37, 39, 99, 111, 117, 143, 297, 333, 351, 407, 429, 481, 999, 1221, 1287, 1443, 3663, 3861, 4329, 5291, 10989, 12987, 15873, 47619, 142857 142,857 is the natural number following 142,856 and preceding 142,858. It is both a Kaprekar number and a cyclic number.https://oeis.org/A006886 Sloane's A006886: Kaprekar numbers The On-Line Encyclopedia of Integer Sequences OEIS Foundation 2016-06-03 Cyclic number 142857 is the best-known cyclic number in base 10, being the six repeating digits of 1 7 (0.142857).http://www.daviddarling.info/encyclopedia/C/cyclic_number.html Cyclic number https://web.archive.org/web/20070929122525/http://www.daviddarling.info/encyclopedia/C/cyclic_number.html 2007-09-29 The Internet Encyclopedia of ScienceMichael W. Ecker 3026586 The Alluring Lore of Cyclic Numbers The Two-Year College Mathematics Journal 14 2 March 1983 105–109 10.2307/3026586 CyclicNumber Cyclic Numberhttp://findarticles.com/p/articles/mi_hb4870/is_200508/ai_n17913296 https://web.archive.org/web/20071224191059/http://findarticles.com/p/articles/mi_hb4870/is_200508/ai_n17913296 usurped 2007-12-24 Go figure (cyclic numbers) Hogan Kathryn Australian Doctor August 2005 If 142857 is multiplied by 2, 3, 4, 5 or 6, the answer will be a cyclic permutation of itself, and will correspond to the repeating digits of 2 7, 3 7, 4 7, 5 7 or 6 7 respectively: 1 × 142,857 = 142,857 2 × 142,857 = 285,714 3 × 142,857 = 428,571 4 × 142,857 = 571,428 5 × 142,857 = 714,285 6 × 142,857 = 857,142 7 × 142,857 = 999,999 If multiplying by an integer greater than 7, there is a simple process to get to a cyclic permutation of 142857. By adding the rightmost six digits (ones through hundred thousands) to the remaining digits and repeating this process until only six digits are left, it will result in a cyclic permutation of 142857:January 2018 142857 × 8 = 1142856 1 + 142856 = 142857 142857 × 815 = 116428455 116 + 428455 = 428571 1428572 = 142857 × 142857 = 20408122449 20408 + 122449 = 142857 Multiplying by a multiple of 7 will result in 999999 through this process: 142857 × 74 = 342999657 342 + 999657 = 999999 If you square the last three digits and subtract the square of the first three digits, you also get back a cyclic permutation of the number.January 2018 8572 = 734449 1422 = 20164 734449 − 20164 = 714285 It is the repeating part in the decimal expansion of the rational number 1 7 = 0.142857. Thus, multiples of 1 7 are simply repeated copies of the corresponding multiples of 142857: Connection to the enneagram The 142857 number sequence is used in the enneagram figure, a symbol of the Gurdjieff Work used to explain and visualize the dynamics of the interaction between the two great laws of the Universe (according to G. I. Gurdjieff), the Law of Three and the Law of Seven. The movement of the numbers of 142857 divided by 1 7, 2 7. etc., and the subsequent movement of the enneagram, are portrayed in Gurdjieff's sacred dances known as the movements.In Search of the Miraculous: Fragments of an Unknown Teaching Ouspensky P. D. Routledge 1947 London Chapter XVIII Other properties The 142857 number sequence is also found in several decimals in which the denominator has a factor of 7. In the examples below, the numerators are all 1, however there are instances where it does not have to be, such as 2 7 (0.285714). For example, consider the fractions and equivalent decimal values listed below: 1 7 = 0.142857... 1 14 = 0.0714285... 1 28 = 0.03571428... 1 35 = 0.0285714... 1 56 = 0.017857142... 1 70 = 0.0142857... The above decimals follow the 142857 rotational sequence. There are fractions in which the denominator has a factor of 7, such as 1 21 and 1 42, that do not follow this sequence and have other values in their decimal digits. References Leslie John The Philosophy of Arithmetic: Exhibiting a Progressive View of the Theory and Practice of… https://archive.org/details/philosophyarith00leslgoog Longman, Hurst, Rees, Orme, and Brown 1820 1-4020-1546-1 Wells D. The Penguin Dictionary of Curious and Interesting Numbers revised London Penguin Group 1997 171–175 978-0-140-26149-3 Tahan Malba The Man Who Counted 1938 10 Category:Integers Category:Numbers