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
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
KEYWORD
nonn,more
AUTHOR
Amiram Eldar, Apr 30 2021
EXTENSIONS
a(18)-a(20) from Martin Ehrenstein, May 15 2021
STATUS
approved