This site is supported by donations to The OEIS Foundation.

 Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing. Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A279048 a(n) = 0 whenever n is a practical number (A005153) otherwise least powers of 2 that when multiplied by n becomes practical. 1

%I

%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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 12 09:36 EST 2019. Contains 329953 sequences. (Running on oeis4.)