Editing Highly cototient number (section)

{{Short description|1 = Numbers k where x - phi(x) = k has many solutions}}

In [[number theory]], a branch of [[mathematics]], a '''highly cototient number''' is a positive [[integer]] <math>k</math> which is above 1 and has more solutions to the [[equation]] 

:<math>x - \phi(x) = k</math>

than any other integer below <math>k</math> and above 1. Here, <math>\phi</math> is [[Euler's totient function]]. There are infinitely many solutions to the equation for 

:<math>k</math> = [[1 (number)|1]] 

so this value is excluded in the definition. The first few highly cototient numbers are:<ref name=a100827>{{Cite OEIS|A100827|name=Highly cototient numbers}}.</ref>

:[[2 (number)|2]], [[4 (number)|4]], [[8 (number)|8]], [[23 (number)|23]], [[35 (number)|35]], [[47 (number)|47]], [[59 (number)|59]], [[63 (number)|63]], [[83 (number)|83]], [[89 (number)|89]], [[113 (number)|113]], [[119 (number)|119]], [[167 (number)|167]], [[209 (number)|209]], [[269 (number)|269]], 299, 329, 389, 419, 509, 629, 659, 779, 839, 1049, 1169, 1259, 1469, 1649, 1679, 1889, ... {{OEIS|id=A100827}}

Many of the highly cototient numbers are odd.<ref name=a100827/>

The concept is somewhat analogous to that of [[highly composite number]]s. Just as there are infinitely many highly composite numbers, there are also infinitely many highly cototient numbers. Computations become harder, since [[integer factorization]] becomes harder as the numbers get larger.