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Cyclic numbers (A003277) which set a record for the gap to the next cyclic number.
1

%I #26 May 16 2021 01:48:38

%S 1,3,7,23,199,2297,3473,124311,262193,580011,2847499,16329689,

%T 115495383,399128719,13657103441,16022594389,66275713667,733100630963,

%U 1291428223783,5340370800707

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

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

%C 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, ...

%H Paul Erdős, <a href="http://www.renyi.hu/~p_erdos/1948-11.pdf">Some asymptotic formulas in number theory</a>, J. Indian Math. Soc. (N.S.), Vol. 12 (1948), pp. 75-78.

%e 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.

%e From _Martin Ehrenstein_, May 11 2021: (Start)

%e Table of the first 4 terms:

%e n | cyclic number | gap

%e ---+---------------+----

%e 1 | 1 | 1

%e | 2 | 1

%e 2 | 3 | 2

%e | 5 | 2

%e 3 | 7 | 4

%e | 11 | 2

%e | 13 | 2

%e | 15 | 2

%e | 17 | 2

%e | 19 | 4

%e 4 | 23 | 6

%e | 29 | ...

%e ...| ... | ...

%e (End)

%t 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

%Y Cf. A003277, A343816.

%K nonn,more

%O 1,2

%A _Amiram Eldar_, Apr 30 2021

%E a(18)-a(20) from _Martin Ehrenstein_, May 15 2021