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%I #68 Feb 03 2024 10:16:10
%S 1,2,4,6,6,10,12,8,16,18,12,22,20,18,28,30,20,24,36,24,40,42,24,46,42,
%T 32,52,40,36,58,60,36,48,66,44,70,72,40,60,78,54,82,64,56,88,72,60,72,
%U 96,60,100,102,48,106,108,72,112,88,72,96,110,80,100,126,84,130
%N a(n) = phi(2n+1).
%C Bisection of A000010 (cf. A062570).
%C From _Alain Rocchelli_, Jun 28 2023: (Start)
%C If 2*n+1 has r distinct odd prime factors, 2^r divides a(n).
%C Conjectures:
%C 1) For any composite integer 2*n+1, a(n) doesn't divide 2*n.
%C 2) For all n, a(n) is never equal to n. (End)
%H T. D. Noe, <a href="/A037225/b037225.txt">Table of n, a(n) for n = 0..10000</a>
%H John Brillhart, J. S. Lomont, and Patrick Morton, <a href="https://eudml.org/doc/151797">Cyclotomic properties of the Rudin-Shapiro polynomials</a>, J. Reine Angew. Math. 288 (1976), 37-65; see Table 2; MR0498479 (58 #16589). - From _N. J. A. Sloane_, Jun 06 2012
%H V. I. Levenshtein, <a href="http://mi.mathnet.ru/eng/ppi/v43/i3/p39">Conflict-avoiding codes and cyclic triple systems</a>, Problemy Peredachi Informatsii, 43 (No. 3, 2007), 39-53 (in Russian).
%H V. I. Levenshtein, <a href="https://doi.org/10.1134/S0032946007030039">Conflict-avoiding codes and cyclic triple systems</a>, Problems of Information. Transmission, 43 (2007), 199-212 (translation from Russian).
%H N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence).
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Lehmer%27s_totient_problem">Lehmer's totient problem</a>.
%F Sum_{k=0..n} a(k) ~ c * n^2, where c = 8/Pi^2 = 0.810569... (A217739). - _Amiram Eldar_, Nov 17 2022
%F a(n) = 2*n iff 2*n+1 is prime, see A005097. - _Alain Rocchelli_, Jun 22 2023
%F From _Peter Bala_, Feb 01 2024: (Start)
%F Odd bisection of A000010.
%F a(n) = 2*A072451(n) for n >= 1.
%F G.f.: Sum_{n >= 1} phi(2*n+1)*x^(2*n+1) = Sum_{n >= 1} moebius(n)*x^(2*n-1)*(1 + x^(4*n-2))/(1 - x^(4*n-2))^2 = x + 2*x^3 + 4*x^5 + 6*x^7 + 6*x^9 + .... (End)
%t Table[EulerPhi[2 n + 1], {n, 0, 80}] (* _Vincenzo Librandi_, Jul 16 2019 *)
%o (Magma) [EulerPhi(2*n+1): n in [0..70]]; // _Vincenzo Librandi_, Jul 16 2019
%o (PARI) a(n) = eulerphi(2*n+1) \\ _Amiram Eldar_, Nov 17 2022
%Y Cf. A000010, A062570, A72451, A217739.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
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