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

1,2

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, ...

Links

Table of n, a(n) for n=1..20.

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

Example

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)

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

Crossrefs

Cf. A003277, A343816.

Sequence in context: A121883 A262264 A060235 * A090188 A001773 A067604

Adjacent sequences: A343812 A343813 A343814 * A343816 A343817 A343818

Keyword

nonn,more

Author

Amiram Eldar, Apr 30 2021

Extensions

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

Status

approved