A279048 - OEIS

%I #19 Jul 25 2017 02:16:45

%S 0,0,1,0,2,0,2,0,1,1,3,0,3,1,1,0,4,0,4,0,1,2,4,0,2,2,1,0,4,0,4,0,1,3,

%T 2,0,5,3,1,0,5,0,5,1,1,3,5,0,2,1,2,1,5,0,2,0,2,3,5,0,5,3,1,0,2,0,6,2,

%U 2,1,6,0,6,4,1,2,2,0,6,0,1,4,6,0,2,4,2,0,6,0,2,2,3,4,2,0,6,1,1,0

%N a(n) = 0 whenever n is a practical number (A005153) otherwise least powers of 2 that when multiplied by n becomes practical.

%C A conjecture by _Zhi-Wei Sun_ states that any rational number can be expressed as the sum of distinct unit fractions whose denominators are practical numbers. To prove this conjecture, _David Eppstein_ (see link) used the fact that every natural number when repeatedly multiplied by 2 will eventually become practical.

%H Vincenzo Librandi, <a href="/A279048/b279048.txt">Table of n, a(n) for n = 1..5000</a>

%H David Eppstein, <a href="https://11011110.github.io/blog/2016/11/20/egyptian-fractions-with.html">Egyptian fractions with practical denominators</a>, Nov 20, 2016

%H Zhi-Wei Sun, <a href="http://maths.nju.edu.cn/~zwsun/UnitFraction.pdf">A conjecture on unit fractions involving primes</a>, preprint, 2015.

%e a(11) = 3 because 11 * 2^3 = 88 is a practical number and 3 is the least power of 2 which when multiplied by 11 becomes practical.

%t practicalQ[n_] := Module[{f, p, e, prod = 1, ok = True}, If[n < 1 ||(n > 1 && OddQ[n]), False, If[n == 1, True, f = FactorInteger[n]; {p, e} = Transpose[f]; Do[If[p[[i]] > 1 + DivisorSigma[1, prod], ok = False; Break[]]; prod = prod * p[[i]]^e[[i]], {i, Length[p]}]; ok]]]; Table[(m = n; k = 0; While[! practicalQ[m], m = 2 * m; k++]; k), {n, 100}]

%Y Cf. A005153.

%K nonn

%O 1,5

%A _Frank M Jackson_, Dec 04 2016