cp's OEIS Frontend

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

%I A387363 #10 Aug 28 2025 10:25:39
%S A387363 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,
%T A387363 13,14,13,16,19,14,19,20,19,20,20,20,21,26,19,24,26,22,25,26,24,26,33,
%U A387363 26,27,30,26,28,32,26,29,38,25,30,34,26,33,34,29,30,41,28
%N A387363 The number of decompositions of 2*n into ordered sums of two cyclic numbers.
%C A387363 Analogous to A002372 with cyclic numbers (A003277) instead of odd primes.
%C A387363 Pomerance (2025) proved that a(n) > 0 for every sufficiently large n.
%H A387363 Amiram Eldar, <a href="/A387363/b387363.txt">Table of n, a(n) for n = 1..10000</a>
%H A387363 Joel E. Cohen, <a href="https://arxiv.org/abs/2508.08335">Conjectures about Primes and Cyclic Numbers</a>, arXiv:2508.08335 [math.NT], 2025.
%H A387363 Carl Pomerance, <a href="https://math.dartmouth.edu/~carlp/cyclic.pdf">Patterns for cyclic numbers</a>, 2025.
%F A387363 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).
%e A387363 a(1) = 1 since 2*1 = 1 + 1.
%e A387363 a(2) = 3 since 2*2 = 1 + 3 = 2 + 2 = 3 + 1.
%e A387363 a(3) = 3 since 2*3 = 1 + 5 = 3 + 3 = 5 + 1.
%t A387363 cyclicQ[n_] := cyclicQ[n] = CoprimeQ[n, EulerPhi[n]]; a[n_] := Count[Range[2*n], _?(And @@ cyclicQ[{#, 2*n-#}] &)]; Array[a, 100]
%o A387363 (PARI) iscyclic(k) = gcd(k, eulerphi(k)) == 1;
%o A387363 a(n) = sum(k = 1, 2*n, iscyclic(k) * iscyclic(2*n-k));
%Y A387363 Cf. A002372, A003277, A387362.
%Y A387363 Cf. A001620, A005597, A073004.
%K A387363 nonn,easy,new
%O A387363 1,2
%A A387363 _Amiram Eldar_, Aug 27 2025