A387363 -id:A387363 - cp's OEIS frontend

A387362 Cyclic numbers k such that k+2 is also a cyclic number.

1, 3, 5, 11, 13, 15, 17, 29, 31, 33, 35, 41, 51, 59, 65, 67, 69, 71, 77, 83, 85, 87, 89, 95, 101, 107, 113, 131, 137, 139, 141, 143, 149, 157, 159, 161, 177, 179, 185, 191, 197, 209, 211, 213, 215, 221, 227, 233, 239, 247, 249, 255, 257, 263, 265, 267, 269, 281, 293

Offset: 1

Amiram Eldar, Aug 27 2025

All the lesser members of twin primes (A001359) are terms since every prime is a cyclic number (A003277).

Cohen (2025) conjectured and Pomerance (2025) proved that this sequence is infinite.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Joel E. Cohen, Conjectures about Primes and Cyclic Numbers, arXiv:2508.08335 [math.NT], 2025.
Carl Pomerance, Patterns for cyclic numbers, 2025.

Cf. A001620, A005597, A073004, A387363.
Subsequence of A003277.
A001359 is a subsequence.

cyclicQ[n_] := cyclicQ[n] = CoprimeQ[n, EulerPhi[n]]; Select[Range[1, 300, 2], And @@ cyclicQ[{#, # + 2}] &]

iscyclic(k) = gcd(k, eulerphi(k)) == 1;
isok(k) = k % 2 && iscyclic(k) && iscyclic(k+2);

The number of terms <= x is ~ 2 * C_2 * x / (exp(gamma) * log(log(log(x))))^2, where C_2 = A005597, and gamma = A001620 (Pomerance, 2025).

cp's OEIS Frontend

A387362 Cyclic numbers k such that k+2 is also a cyclic number.

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