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
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.
Programs
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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));
Comments