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A349128 a(n) = phi(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and phi is Euler totient function.

%I A349128 #18 Dec 24 2022 03:49:44
%S A349128 1,1,1,1,2,1,4,1,2,2,6,1,10,4,2,1,12,2,16,2,4,6,18,1,6,10,4,4,22,2,28,
%T A349128 1,6,12,8,2,30,16,10,2,36,4,40,6,4,18,42,1,20,6,12,10,46,4,12,4,16,22,
%U A349128 52,2,58,28,8,1,20,6,60,12,18,8,66,2,70,30,6,16,24,10,72,2,8,36,78,4,24,40,22,6,82,4,40
%N A349128 a(n) = phi(A064989(n)), where A064989 is multiplicative with a(2^e) = 1 and a(p^e) = prevprime(p)^e for odd primes p, and phi is Euler totient function.
%C A349128 See comments in A349127.
%H A349128 Antti Karttunen, <a href="/A349128/b349128.txt">Table of n, a(n) for n = 1..20000</a>
%H A349128 <a href="/index/Pri#prime_indices">Index entries for sequences computed from indices in prime factorization</a>.
%F A349128 Multiplicative with a(2^e) = 1, and for odd primes p, a(p^e) = (q - 1)*q^(e-1), where q = prevprime(p), where prevprime is A151799.
%F A349128 For odd n, a(n) = A349127(n), for even n, a(n) = a(n/2).
%F A349128 For all n >= 1, a(n) = a(2*n) = a(A000265(n)).
%F A349128 For all n >= 1, a(A000040(1+n)) = A006093(n) = A000040(n)-1.
%F A349128 Sum_{k=1..n} a(k) ~ c * n^2, where c = (64/(3*Pi^4)) / Product_{p prime > 2} (1+1/p-q(p)/p^2-q(p)/p^3) = 0.17889586..., where q(p) = prevprime(p) = A151799(p). - _Amiram Eldar_, Dec 24 2022
%t A349128 f[p_, e_] := If[p == 2, 1, Module[{q = NextPrime[p, -1]}, (q - 1)*q^(e - 1)]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 24 2022 *)
%o A349128 (PARI) A349128(n) = { my(f = factor(n), q); prod(i=1, #f~, if(2==f[i,1], 1, q = precprime(f[i,1]-1); (q-1)*(q^(f[i,2]-1)))); };
%Y A349128 Agrees with A347115, A348045 and A349127 on odd numbers.
%Y A349128 Cf. A285702 (odd bisection).
%Y A349128 Cf. A000010, A064989, A151799, A349122 (inverse Möbius transform).
%Y A349128 Cf. also A003972.
%K A349128 nonn,mult
%O A349128 1,5
%A A349128 _Antti Karttunen_, Nov 13 2021