A080736 Multiplicative function defined by a(1)=1, a(2)=0, a(2^r) = phi(2^r) (r>1), a(p^r) = phi(p^r) (p odd prime, r>=1), where phi is Euler's function A000010.
1, 0, 2, 2, 4, 0, 6, 4, 6, 0, 10, 4, 12, 0, 8, 8, 16, 0, 18, 8, 12, 0, 22, 8, 20, 0, 18, 12, 28, 0, 30, 16, 20, 0, 24, 12, 36, 0, 24, 16, 40, 0, 42, 20, 24, 0, 46, 16, 42, 0, 32, 24, 52, 0, 40, 24, 36, 0, 58, 16, 60, 0, 36, 32, 48, 0, 66, 32, 44, 0, 70, 24, 72, 0, 40, 36, 60, 0, 78, 32
Offset: 1
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Programs
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Haskell
a080736 n = if n `mod` 4 == 2 then 0 else a000010 n -- Reinhard Zumkeller, Jun 13 2012, Jun 11 2012
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Mathematica
a[n_] := If[Mod[n, 4] == 2, 0, EulerPhi[n]]; Array[a, 100] (* Amiram Eldar, Nov 02 2023 *)
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PARI
{for(n=1,81,f=factor(n); print1(if(n==1,1,if(f[1,1]==2&&f[1,2]==1,0,prod(j=1,matsize(f)[1],eulerphi(f[j,1]^f[j,2])))),","))}
Formula
a(n) = if n mod 4 = 2 then 0 else A000010(n). - Reinhard Zumkeller, Jun 13 2012
From Amiram Eldar, Nov 02 2023: (Start)
Multiplicative with a(2) = 0, a(2^e) = 2^(e-1) for e >= 2, and a(p^e) = (p-1)*p^(e-1) for an odd prime p.
Dirichlet g.f.: (1 - 2^(1-s) + 1/(2^s-1)) * zeta(s-1) / zeta(s).
Sum_{k=1..n} a(k) ~ (5/(2*Pi^2)) * n^2. (End)
Extensions
More terms and PARI code from Klaus Brockhaus, Mar 10 2003