A179651 Difference between consecutive practical numbers.

1, 2, 2, 2, 4, 4, 2, 2, 4, 4, 2, 2, 4, 4, 2, 6, 6, 2, 4, 4, 2, 6, 6, 2, 4, 4, 2, 6, 4, 4, 4, 4, 8, 6, 2, 4, 8, 4, 6, 6, 4, 2, 6, 8, 4, 12, 4, 2, 2, 4, 4, 2, 6, 4, 4, 4, 6, 6, 12, 4, 4, 4, 6, 2, 4, 4, 8, 6, 6, 4, 2, 2, 4, 8, 4, 6, 6, 4, 2, 6, 4, 8, 4, 4, 10, 2, 4, 6, 2, 4, 4, 8, 6, 2, 4, 12, 8, 8, 2, 6, 4, 2, 2

Offset: 1

Author

Jason G. Wurtzel, Jul 22 2010

Keywords

• easy
• nonn

Comments

Because the density of practical numbers is comparable to that of primes, it is natural to inquire whether certain results about prime numbers and their gaps carry over to practical numbers and their gaps. For example, it is known that lim inf a(n) = 2, which is comparable to the twin prime conjecture; and since the density of the practical numbers is zero, it follows that a(n) is unbounded. - Hal M. Switkay, Jan 21 2023

Examples

			For n=3, this is 6-4=2.
For n=5, this is 12-8=4.
		

Links

• Hal M. Switkay, Table of n, a(n) for n = 1..9999

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]]]; s = Select[ Range@ 479, PracticalQ]; Rest@s - Most@s (* Robert G. Wilson v, Jul 23 2010 *)

Formula

a(n) = A005153(n+1) - A005153(n).

Extensions

a(20) onwards from Robert G. Wilson v, Jul 23 2010