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

Author

Amiram Eldar, Aug 27 2025

Keywords

• nonn
• easy
• new

Comments

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

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

Examples

			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.
		

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.

Crossrefs

Cf. A002372, A003277, A387362.

Cf. A001620, A005597, A073004.

Programs

Mathematica

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

PARI

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

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).