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A349136 Möbius transform of Kimberling's paraphrases, A003602.

%I A349136 #29 Nov 24 2023 12:38:16
%S A349136 1,0,1,0,2,0,3,0,3,0,5,0,6,0,4,0,8,0,9,0,6,0,11,0,10,0,9,0,14,0,15,0,
%T A349136 10,0,12,0,18,0,12,0,20,0,21,0,12,0,23,0,21,0,16,0,26,0,20,0,18,0,29,
%U A349136 0,30,0,18,0,24,0,33,0,22,0,35,0,36,0,20,0,30,0,39,0,27,0,41,0,32,0,28,0,44,0,36,0,30,0,36
%N A349136 Möbius transform of Kimberling's paraphrases, A003602.
%H A349136 Antti Karttunen, <a href="/A349136/b349136.txt">Table of n, a(n) for n = 1..20000</a>
%F A349136 a(n) = Sum_{d|n} A008683(d) * A003602(n/d).
%F A349136 a(1) = 1, a(n) = A000010(n)/2 for odd n > 1, a(n) = 0 for even n.
%F A349136 For all n >= 1, a(2*n-1) = A055034(2*n-1) = A072451(n).
%F A349136 a(n) = phi(n) - (1/2)*phi(2n), for n>1. - _Ridouane Oudra_, Jul 13 2023
%F A349136 Sum_{k=1..n} a(k) ~ (1/Pi^2)*n^2. - _Amiram Eldar_, Jul 15 2023
%p A349136 with(numtheory): a:=proc(n) if n=1 then 1; elif n mod 2 = 0 then 0; else phi(n)/2; fi: end proc: seq(a(n), n=1..60); # _Ridouane Oudra_, Jul 13 2023
%t A349136 k[n_] := (n/2^IntegerExponent[n, 2] + 1)/2; a[n_] := DivisorSum[n, MoebiusMu[#] * k[n/#] &]; Array[a, 100] (* _Amiram Eldar_, Nov 13 2021 *)
%o A349136 (PARI) A349136(n) = if(1==n,1, if(n%2, eulerphi(n)/2, 0));
%o A349136 (PARI)
%o A349136 A003602(n) = (1+(n>>valuation(n,2)))/2;
%o A349136 A349136(n) = sumdiv(n,d,moebius(d)*A003602(n/d));
%o A349136 (Python)
%o A349136 from sympy import totient
%o A349136 def A349136(n): return totient(n)+1>>1 if n&1 else 0 # _Chai Wah Wu_, Nov 24 2023
%Y A349136 Agrees with A055034 on odd arguments.
%Y A349136 Cf. A000010, A003602, A008683, A349134.
%Y A349136 Cf. A000004, A072451 (even and odd bisection).
%Y A349136 Cf. also A347233, A349127, A349137.
%K A349136 nonn
%O A349136 1,5
%A A349136 _Antti Karttunen_, Nov 13 2021