A068773 Alternating sum phi(1) - phi(2) + phi(3) - phi(4) + ... + ((-1)^(n+1))*phi(n).
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, 80, 68, 96, 88, 118, 102, 122, 106, 130, 118, 154, 136, 160, 144, 184, 172, 214, 194, 218, 196, 242, 226, 268, 248, 280, 256, 308, 290, 330, 306, 342, 314, 372, 356
Offset: 1
Examples
a(3) = phi(1) - phi(2) + phi(3) = 1 - 1 + 2 = 2.
Links
- Amiram Eldar, Table of n, a(n) for n = 1..10000
- László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.
Programs
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Maple
with(numtheory): seq(add((-1)^(k+1)*phi(k),k=1..n), n=1..80); # Ridouane Oudra, Mar 22 2024
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Mathematica
Accumulate[Array[(-1)^(# + 1) * EulerPhi[#] &, 100]] (* Amiram Eldar, Oct 14 2022 *)
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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,","))
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Python
# uses code from A002088 and A049690 def A068773(n): return A002088(n)-(A049690(n>>1)<<1) # Chai Wah Wu, Aug 04 2024
Formula
a(n) = Sum_{k=1..n} (-1)^(k+1)*phi(k).
a(n) = n^2/Pi^2 + O(n * log(n)^(2/3) * log(log(n))^(4/3)) (Tóth, 2017). - Amiram Eldar, Oct 14 2022