A343815 Cyclic numbers (A003277) which set a record for the gap to the next cyclic number.

1, 3, 7, 23, 199, 2297, 3473, 124311, 262193, 580011, 2847499, 16329689, 115495383, 399128719, 13657103441, 16022594389, 66275713667, 733100630963, 1291428223783, 5340370800707

Offset: 1

Author

Amiram Eldar, Apr 30 2021

Keywords

• nonn
• more

Comments

Since the asymptotic density of the cyclic numbers is 0 (Erdős, 1948), this sequence is infinite.

The corresponding record values are 1, 2, 4, 6, 10, 12, 14, 16, 18, 20, 22, 24, 26, 30, 32, 34, 36, 38, 40, 42, ...

Examples

			The first 6 cyclic numbers are 1, 2, 3, 5, 7 and 11. The gaps between them are 1, 1, 2, 2 and 4. The record gaps, 1, 2 and 4, occur after the cyclic numbers 1, 3 and 7, which are the first 3 terms of this sequence.
From _Martin Ehrenstein_, May 11 2021: (Start)
Table of the first 4 terms:
  n  | cyclic number | gap
  ---+---------------+----
  1  |       1       |  1
     |       2       |  1
  2  |       3       |  2
     |       5       |  2
  3  |       7       |  4
     |      11       |  2
     |      13       |  2
     |      15       |  2
     |      17       |  2
     |      19       |  4
  4  |      23       |  6
     |      29       | ...
  ...|     ...       | ...
(End)
		

Links

• Paul Erdős, Some asymptotic formulas in number theory, J. Indian Math. Soc. (N.S.), Vol. 12 (1948), pp. 75-78.

Crossrefs

Cf. A003277, A343816.

Programs

Mathematica

cycQ[n_] := CoprimeQ[n, EulerPhi[n]]; seq = {}; m = 1; dm = 0; Do[If[cycQ[n], d = n - m; If[d > dm, dm = d; AppendTo[seq, m]]; m = n], {n, 2, 10^6}]; seq

Extensions

a(18)-a(20) from Martin Ehrenstein, May 15 2021