A387363 The number of decompositions of 2*n into ordered sums of two cyclic numbers.

1, 3, 3, 4, 3, 4, 5, 6, 8, 8, 7, 8, 7, 6, 9, 8, 11, 12, 11, 10, 12, 12, 13, 16, 12, 14, 16, 12, 13, 14, 13, 16, 19, 14, 19, 20, 19, 20, 20, 20, 21, 26, 19, 24, 26, 22, 25, 26, 24, 26, 33, 26, 27, 30, 26, 28, 32, 26, 29, 38, 25, 30, 34, 26, 33, 34, 29, 30, 41, 28

Offset

1,2

Comments

Analogous to A002372 with cyclic numbers (A003277) instead of odd primes.

Pomerance (2025) proved that a(n) > 0 for every sufficiently large n.

Links

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.

Formula

a(n) ~ C_2 * n / (exp(gamma) * log(log(log(n))))^2 * Product_{p | n, p odd prime < log(log(n/2))} (p-1)/(p-2), where C_2 = A005597, and gamma = A001620 (Pomerance, 2025).

Example

a(1) = 1 since 2*1 = 1 + 1.
a(2) = 3 since 2*2 = 1 + 3 = 2 + 2 = 3 + 1.
a(3) = 3 since 2*3 = 1 + 5 = 3 + 3 = 5 + 1.

Mathematica

cyclicQ[n_] := cyclicQ[n] = CoprimeQ[n, EulerPhi[n]]; a[n_] := Count[Range[2*n], _?(And @@ cyclicQ[{#, 2*n-#}] &)]; Array[a, 100]

Prog

(PARI) iscyclic(k) = gcd(k, eulerphi(k)) == 1;
a(n) = sum(k = 1, 2*n, iscyclic(k) * iscyclic(2*n-k));

Crossrefs

Cf. A002372, A003277, A387362.

Cf. A001620, A005597, A073004.

Sequence in context: A175928 A161357 A138372 * A316662 A123708 A350501

Adjacent sequences: A387360 A387361 A387362 * A387364 A387365 A387366

Keyword

nonn,easy

Author

Amiram Eldar, Aug 27 2025

Status

approved