OFFSET
1,2
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
LINKS
Hal M. Switkay, Table of n, a(n) for n = 1..9999
EXAMPLE
For n=3, this is 6-4=2.
For n=5, this is 12-8=4.
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 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jason G. Wurtzel, Jul 22 2010
EXTENSIONS
a(20) onwards from Robert G. Wilson v, Jul 23 2010
STATUS
approved