# Probability distribution of count of factors for all numbers

Is the following known?

Define "factor count" as the number of prime factors of the number, minus 1. For example:

Prime numbers have a factor count of 1-1 = 0 4 has a factor count of (2 and 2)-1 = 1 20 has a factor count of (2 and 2 and 5)-1 = 2 24 has a factor count of (2 and 2 and 2 and 3)-1 = 3 etc.

When you plot the factor counts of all numbers, it becomes a Poisson distribution with $\lambda=e$. I have written a program that shows this.

## 1 Answer

Actually it does become essentially normally distributed (like the Poisson distribution), but the mean is not $e$ but rather $\log\log n.$ This was discovered by Erdős and Kac in the late 30s.

By way of demonstration, consider the numbers around a googol ($10^{100}$) which have $\log\log n\approx5.4$:

10^100+1 = 73 * 137 * 401 * 1201 * 1601 * 1676321 * 5964848081 * 129694419029057750551385771184564274499075700947656757821537291527196801
10^100+2 = 2 * 3 * 4832936419 * 5025493293281 * 1061431139892014340488875721 * 64649794020110132416875748306224068640129784020593
10^100+3 = 7 * 157 * 769 * 2593 * 4888946572366141 * 220030935994058489226133 * 4242036622639156527888055237578804493024993216233097
10^100+4 = 2^2 * 20794121 * 319929089 * 406288107529 * 5918277534160279189665941011889 * 156284632186102964835435736198404890903809
10^100+5 = 3 * 5 * 127 * 570527 * 9200868376117164262739684518876954703209497971776676688390514959123739106672139895228883723
10^100+6 = 2 * 859493 * 2698836149 * 46606393157 * 201991350982876187 * 6930035321787863868408416051 * 33039801179985499802003182831
10^100+7 = 557 * 294001 * 6908913964859 * 8838655616713081384235006409051132181948832801178384207854273111302870957443689
10^100+8 = 2^3 * 3^2 * 113 * 593 * 48673 * 8181960160259 * 3293045699351804081581417551013427 * 1580487747467038622482181403711428970986689
10^100+9 = 3221 * 426362206609 * 7281662972128939980921782529252917011318952210150992083439865279439412269153282637781
10^100+10 = 2 * 5 * 7 * 11^2 * 13 * 19 * 23 * 4093 * 8779 * 52579 * 599144041 * 7093127053 * 183411838171 * 141122524877886182282233539317796144938305111168717


These have (what you call) factor counts of 7, 5, 6, 6, 4, 6, 3, 10, 2, and 14. Does those look closer to 2.7 or 5.4?