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A179651
Difference between consecutive practical numbers.
3
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
1,2
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
FORMULA
a(n) = A005153(n+1) - A005153(n).
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
Cf. A005153.
Sequence in context: A367120 A064135 A332476 * A139560 A192095 A363710
KEYWORD
easy,nonn
AUTHOR
Jason G. Wurtzel, Jul 22 2010
EXTENSIONS
a(20) onwards from Robert G. Wilson v, Jul 23 2010
STATUS
approved