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A353748 a(n) = phi(n) - A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.

%I A353748 #17 Dec 21 2023 10:22:39
%S A353748 0,0,0,1,1,0,1,3,2,1,3,2,1,1,2,7,3,2,1,5,2,3,3,6,11,1,10,7,5,2,1,15,6,
%T A353748 3,9,8,5,1,2,13,3,2,1,13,12,3,3,14,17,11,6,13,5,10,19,19,2,5,5,10,1,1,
%U A353748 16,31,15,6,5,19,6,9,3,20,1,5,22,19,25,2,5,29,38,3,3,14,25,1,10,33,5,12,17,25,2,3,21
%N A353748 a(n) = phi(n) - A064989(n), where phi is Euler totient function, and A064989 shifts the prime factorization one step towards lower primes.
%H A353748 Antti Karttunen, <a href="/A353748/b353748.txt">Table of n, a(n) for n = 1..16384</a>
%H A353748 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%F A353748 a(n) = A000010(n) - A064989(n).
%F A353748 For n >= 0, a(4n+2) = a(2n+1).
%F A353748 For n > 1, a(n) = A140434(n) - A353747(n).
%F A353748 Sum_{k=1..n} a(k) ~ c * n^2, where c = 3/Pi^2 - (1/2) * Product_{p prime} ((p^2-p)/(p^2-q(p))) = 0.0832596219... , where q(p) = prevprime(p) (A151799) if p > 2 and q(2) = 1. - _Amiram Eldar_, Dec 21 2023
%t A353748 f1[p_, e_] := (p - 1)*p^(e - 1); f2[p_, e_] := If[p == 2, 1, NextPrime[p, -1]^e]; a[1] = 0; a[n_] := Times @@ f1 @@@ (f = FactorInteger[n]) - Times @@ f2 @@@ f; Array[a, 100] (* _Amiram Eldar_, May 07 2022 *)
%o A353748 (PARI)
%o A353748 A064989(n) = { my(f=factor(n>>valuation(n,2))); for(i=1, #f~, f[i,1] = precprime(f[i,1]-1)); factorback(f); };
%o A353748 A353748(n) = (eulerphi(n)-A064989(n));
%Y A353748 Cf. A000010, A064989, A140434, A151799, A353747.
%K A353748 nonn
%O A353748 1,8
%A A353748 _Antti Karttunen_, May 06 2022