A279048 a(n) = 0 whenever n is a practical number (A005153) otherwise least powers of 2 that when multiplied by n becomes practical.

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, 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, 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

Offset: 1

Author

Frank M Jackson, Dec 04 2016

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• David Eppstein, Egyptian fractions with practical denominators, Nov 20, 2016
• Zhi-Wei Sun, A conjecture on unit fractions involving primes, preprint, 2015.

Crossrefs

Cf. A005153.

Programs

Mathematica

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}]