which approaches quickly one with increasing ''p''. These numbers clusters on the line of slope 1 in the plots of <math> \varphi(n) </math> and along the top horizontal line in the plots for <math> \varphi(n)/n </math>.
Now many numbers <math> n </math> factor with a few small prime factors and a single much larger prime number. likeA few examples are <math> 8856995 = 5\times 7 \times 36151 </math>, <math> 9502803 = 3^2 \times 1,055,867</math>, <math> 9098763 = 3 \times 3,032,921</math> or a last one <math> 9759450 = 2 \times 3 \times 5^2 \times 65,063</math>. All these numbers yields for <math> \frac{\varphi(n)}{n} </math> simple fractions like <math> 4/5\times 6/7, 2/3, 1/2\times 2/3 \times 4/5 </math> multiplied by a fraction <math> (p-1)/p \approx 1 </math> where <math> p </math> is a large prime. These fractions are seen in the plots as near horizontal lines with large concentration of numbers or as vertical lines in the spectrum-like plot. The list of small primes originating the pattern is indicated on the right of the horizontal line or on the top of the corresponding spectral lines. For example <math> \{3,7, p \gg 1\} </math> means a number with prime decomposition as <math> n = 3^\alpha 7^\beta p </math> where <math> p </math> is a large prime. In these cases we have with a good approximation
