A037225 a(n) = phi(2n+1).

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, 32, 52, 40, 36, 58, 60, 36, 48, 66, 44, 70, 72, 40, 60, 78, 54, 82, 64, 56, 88, 72, 60, 72, 96, 60, 100, 102, 48, 106, 108, 72, 112, 88, 72, 96, 110, 80, 100, 126, 84, 130

Offset

0,2

Comments

Bisection of A000010 (cf. A062570).

From Alain Rocchelli, Jun 28 2023: (Start)

If 2*n+1 has r distinct odd prime factors, 2^r divides a(n).

Conjectures:

1) For any composite integer 2*n+1, a(n) doesn't divide 2*n.

2) For all n, a(n) is never equal to n. (End)

Links

T. D. Noe, Table of n, a(n) for n = 0..10000

John Brillhart, J. S. Lomont, and Patrick Morton, Cyclotomic properties of the Rudin-Shapiro polynomials, J. Reine Angew. Math. 288 (1976), 37-65; see Table 2; MR0498479 (58 #16589). - From N. J. A. Sloane, Jun 06 2012

V. I. Levenshtein, Conflict-avoiding codes and cyclic triple systems, Problemy Peredachi Informatsii, 43 (No. 3, 2007), 39-53 (in Russian).

V. I. Levenshtein, Conflict-avoiding codes and cyclic triple systems, Problems of Information. Transmission, 43 (2007), 199-212 (translation from Russian).

N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence).

Wikipedia, Lehmer's totient problem.

Formula

Sum_{k=0..n} a(k) ~ c * n^2, where c = 8/Pi^2 = 0.810569... (A217739). - Amiram Eldar, Nov 17 2022

a(n) = 2*n iff 2*n+1 is prime, see A005097. - Alain Rocchelli, Jun 22 2023

From Peter Bala, Feb 01 2024: (Start)

Odd bisection of A000010.

a(n) = 2*A072451(n) for n >= 1.

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)

Mathematica

Table[EulerPhi[2 n + 1], {n, 0, 80}] (* Vincenzo Librandi, Jul 16 2019 *)

Prog

(Magma) [EulerPhi(2*n+1): n in [0..70]]; // Vincenzo Librandi, Jul 16 2019
(PARI) a(n) = eulerphi(2*n+1) \\ Amiram Eldar, Nov 17 2022

Crossrefs

Cf. A000010, A062570, A072451, A217739.

Sequence in context: A070229 A248835 A053568 * A060685 A073353 A254734

Adjacent sequences: A037222 A037223 A037224 * A037226 A037227 A037228

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved