A343817 Refactorable numbers (A033950) which set a record for the gap to the next refactorable number.

1, 2, 24, 40, 108, 156, 296, 732, 1692, 31616, 51608, 568720, 766620, 6195132, 6938752, 17879440, 18578320, 35196584, 228694176, 475292728, 589169184, 1451254356, 3252050592, 4865544096, 6328305120, 8082626976, 8694028264, 9112984448, 30328732568, 46093418640

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

Table of n, a(n) for n=1..30.

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

Cf. A033950, A114617.

Sequence in context: A119066 A019996 A368042 * A119070 A181283 A368360

Adjacent sequences: A343814 A343815 A343816 * A343818 A343819 A343820

Keyword

nonn

Author

Amiram Eldar, Apr 30 2021

Status

approved