A179651 - OEIS

%I #20 Jun 29 2023 09:06:40

%S 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,

%T 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,

%U 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

%N Difference between consecutive practical numbers.

%C 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

%H Hal M. Switkay, <a href="/A179651/b179651.txt">Table of n, a(n) for n = 1..9999</a>

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

%e For n=3, this is 6-4=2.

%e For n=5, this is 12-8=4.

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

%Y Cf. A005153.

%K easy,nonn

%O 1,2

%A _Jason G. Wurtzel_, Jul 22 2010

%E a(20) onwards from _Robert G. Wilson v_, Jul 23 2010