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Alternating sum phi(1) - phi(2) + phi(3) - phi(4) + ... + ((-1)^(n+1))*phi(n).
16

%I #26 Aug 04 2024 20:36:09

%S 1,0,2,0,4,2,8,4,10,6,16,12,24,18,26,18,34,28,46,38,50,40,62,54,74,62,

%T 80,68,96,88,118,102,122,106,130,118,154,136,160,144,184,172,214,194,

%U 218,196,242,226,268,248,280,256,308,290,330,306,342,314,372,356

%N Alternating sum phi(1) - phi(2) + phi(3) - phi(4) + ... + ((-1)^(n+1))*phi(n).

%H Amiram Eldar, <a href="/A068773/b068773.txt">Table of n, a(n) for n = 1..10000</a>

%H László Tóth, <a href="https://www.emis.de/journals/JIS/VOL20/Toth/toth25.html">Alternating Sums Concerning Multiplicative Arithmetic Functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

%F a(n) = Sum_{k=1..n} (-1)^(k+1)*phi(k).

%F a(n) = n^2/Pi^2 + O(n * log(n)^(2/3) * log(log(n))^(4/3)) (Tóth, 2017). - _Amiram Eldar_, Oct 14 2022

%F a(n) = 3*A002088(n) - 2*A049690(n). - _Ridouane Oudra_, Mar 22 2024

%F a(n) = A002088(n) - 2*A049690(floor(n/2)). - _Chai Wah Wu_, Aug 04 2024

%e a(3) = phi(1) - phi(2) + phi(3) = 1 - 1 + 2 = 2.

%p with(numtheory): seq(add((-1)^(k+1)*phi(k),k=1..n), n=1..80); # _Ridouane Oudra_, Mar 22 2024

%t Accumulate[Array[(-1)^(# + 1) * EulerPhi[#] &, 100]] (* _Amiram Eldar_, Oct 14 2022 *)

%o (PARI) a068773(m)=local(s,n); s=0; for(n=1,m, if(n%2==0,s=s-eulerphi(n),s=s+eulerphi(n)); print1(s,","))

%o (Python)

%o # uses code from A002088 and A049690

%o def A068773(n): return A002088(n)-(A049690(n>>1)<<1) # _Chai Wah Wu_, Aug 04 2024

%Y Cf. A000010, A067929.

%Y Cf. A002088, A049690.

%K easy,nonn

%O 1,3

%A _Klaus Brockhaus_, Feb 28 2002