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

1,5

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.

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.

Example

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.

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}]

Crossrefs

Cf. A005153.

Sequence in context: A282432 A046922 A193779 * A263485 A263489 A238660

Adjacent sequences: A279045 A279046 A279047 * A279049 A279050 A279051

Keyword

nonn

Author

Frank M Jackson, Dec 04 2016

Status

approved