OFFSET
1,2
COMMENTS
Since the asymptotic density of the refactorable numbers is 0 (Kennedy and Cooper, 1990), this sequence is infinite.
The corresponding record values are 1, 6, 12, 16, 20, 24, 32, 44, 92, 100, 144, 152, 180, 192, 208, 212, 236, 268, 280, 296, 336, 360, 368, 372, 384, 396, 408, 432, 488, 496, ...
LINKS
Robert E. Kennedy and Curtis N. Cooper, Tau numbers, natural density and Hardy and Wright's Theorem 437, International Journal of Mathematics and Mathematical Sciences, Vol. 13, No. 2 (1990), pp. 383-386.
EXAMPLE
The first 8 refactorable numbers are 1, 2, 8, 9, 12, 18, 24 and 36. The gaps between them are 1, 6, 1, 3, 6, 6 and 12. The record gaps, 1, 6 and 12, occur after the refactorable numbers 1, 2 and 24, which are the first 3 terms of this sequence.
MATHEMATICA
refQ[n_] := Divisible[n, DivisorSigma[0, n]]; seq = {}; m = 1; dm = 0; Do[If[refQ[n], d = n - m; If[d > dm, dm = d; AppendTo[seq, m]]; m = n], {n, 2, 10^6}]; seq
CROSSREFS
KEYWORD
nonn
AUTHOR
Amiram Eldar, Apr 30 2021
STATUS
approved