cp's OEIS Frontend

A062570 a(n) = phi(2*n).

%I A062570 #107 May 18 2025 03:17:06
%S A062570 1,2,2,4,4,4,6,8,6,8,10,8,12,12,8,16,16,12,18,16,12,20,22,16,20,24,18,
%T A062570 24,28,16,30,32,20,32,24,24,36,36,24,32,40,24,42,40,24,44,46,32,42,40,
%U A062570 32,48,52,36,40,48,36,56,58,32,60,60,36,64,48,40,66,64,44,48,70,48,72
%N A062570 a(n) = phi(2*n).
%C A062570 a(n) is also the number of non-congruent solutions to x^2 - y^2 == 1 (mod n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 21 2003
%C A062570 a(n) is the size of a square companion matrix of the minimal cyclotomic polynomial of (-1)^(1/n). - _Eric Desbiaux_, Dec 08 2015
%C A062570 a(n) is the degree of the (2n)-th cyclotomic field Q(zeta_(2n)). Note that Q(zeta_n) = Q(zeta_(2n)) for odd n. - _Jianing Song_, May 17 2021
%C A062570 The number of integers k from 1 to n such that gcd(n,k) is a power of 2. - _Amiram Eldar_, May 18 2025
%D A062570 T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, p. 28.
%H A062570 Antti Karttunen, <a href="/A062570/b062570.txt">Table of n, a(n) for n = 1..65537</a> (first 1000 terms from Vincenzo Librandi)
%H A062570 N. J. A. Sloane, <a href="/A115004/a115004.txt">Families of Essentially Identical Sequences</a>, Mar 24 2021 (Includes this sequence).
%H A062570 László Tóth, <a href="http://arxiv.org/abs/1404.4214">Counting solutions of quadratic congruences in several variables revisited</a>, arXiv preprint arXiv:1404.4214 [math.NT], 2014.
%H A062570 László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Toth/toth12.html">Counting Solutions of Quadratic Congruences in Several Variables Revisited</a>, J. Int. Seq. 17 (2014), Article 14.11.6.
%H A062570 Wikipedia, <a href="http://en.wikipedia.org/wiki/Ramanujan&#39;s_sum">Ramanujan's sum</a>.
%F A062570 a(n) = Sum_{d|n and d is odd} n/d*mu(d).
%F A062570 Multiplicative with a(2^e) = 2^e and a(p^e) = p^e-p^(e-1), p>2.
%F A062570 Dirichlet g.f.: zeta(s-1)/zeta(s)*2^s/(2^s-1). - _Ralf Stephan_, Jun 17 2007
%F A062570 a(n) = A000010(2*n).
%F A062570 a(n) = phi(n)*(1+((n+1) mod 2)). - _Gary Detlefs_, Jul 13 2011
%F A062570 a(n) = A173557(n)*b(n) where b(n) = 1, 2, 1, 4, 1, 2, 1, 8, 3, 2, 1, 4, 1, 2, ... is the multiplicative function defined by b(p^e) = p^(e-1) if p<>2 and b(2^e)=2^e. b(n) = n/A204455(n). - _R. J. Mathar_, Jul 02 2013
%F A062570 a(n) = -c_{2n}(n) where c_q(n) is Ramanujan's sum. - _Michael Somos_, Aug 23 2013
%F A062570 a(n) = A055034(2*n), for n >= 2. - _Wolfdieter Lang_, Nov 30 2013
%F A062570 O.g.f.: Sum_{n >= 1} mu(2*n-1)*x^(2*n-1)/(1 - x^(2*n-1))^2. - _Peter Bala_, Mar 17 2019
%F A062570 a(n) = A000010(4*n)/2, for n > = 1 (see Apostol, Theorem 2.5, (b), p. 28). - _Wolfdieter Lang_, Nov 17 2019
%F A062570 a(n) = n - Sum_{d|n, n/d odd, d < n} a(d). - _Ilya Gutkovskiy_, May 30 2020
%F A062570 Dirichlet convolution of A000010 and A209229. - _Werner Schulte_, Jan 17 2021
%F A062570 From _Richard L. Ollerton_, May 07 2021: (Start)
%F A062570 a(n) = Sum_{k=1..n} A209229(gcd(n,k)).
%F A062570 a(n) = Sum_{k=1..n} A209229(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
%F A062570 Sum_{k=1..n} a(k) ~ c * n^2, where c = 4/Pi^2 = 0.405284... (A185199). - _Amiram Eldar_, Oct 22 2022
%F A062570 a(n) = A000034(n) * A000010(n). - _Amiram Eldar_, May 18 2025
%p A062570 [phi(2*n)$n=1..80]; # _Muniru A Asiru_, Mar 18 2019
%t A062570 Table[EulerPhi[2 n], {n, 80}] (* _Vincenzo Librandi_, Aug 23 2013 *)
%o A062570 (PARI) a(n) = eulerphi(2*n)
%o A062570 (Sage) [euler_phi(2*n) for n in range(1,74)] # _Zerinvary Lajos_, Jun 06 2009
%o A062570 (Python)
%o A062570 from sympy import totient
%o A062570 def A062570(n): return totient(n) if n&1 else totient(n)<<1 # _Chai Wah Wu_, Aug 04 2024
%Y A062570 Cf. A000010, A000034, A008683, A037225, A060968, A062803, A185199, A209229.
%Y A062570 Column 1 of A129559, column 2 of A372673.
%Y A062570 Row 1 of A379010.
%Y A062570 Row sums of A129558 and of A129564.
%K A062570 mult,nonn,easy
%O A062570 1,2
%A A062570 _Jason Earls_, Jul 03 2001
%E A062570 Corrected by _Vladeta Jovovic_, Dec 04 2002