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A378220 Dirichlet inverse of phi(A003961(n)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).

%I A378220 #20 Jan 13 2025 01:39:05
%S A378220 1,-2,-4,-2,-6,8,-10,-2,-4,12,-12,8,-16,20,24,-2,-18,8,-22,12,40,24,
%T A378220 -28,8,-6,32,-4,20,-30,-48,-36,-2,48,36,60,8,-40,44,64,12,-42,-80,-46,
%U A378220 24,24,56,-52,8,-10,12,72,32,-58,8,72,20,88,60,-60,-48,-66,72,40,-2,96,-96,-70,36,112,-120,-72,8,-78,80,24
%N A378220 Dirichlet inverse of phi(A003961(n)), where A003961 is fully multiplicative function with a(prime(i)) = prime(i+1).
%H A378220 Antti Karttunen, <a href="/A378220/b378220.txt">Table of n, a(n) for n = 1..20000</a>
%H A378220 <a href="/index/Pri#prime_indices">Index entries for sequences related to prime indices in the factorization of n</a>.
%F A378220 Multiplicative with a(p^e) = (1-q), where q = A151800(p), i.e., the least prime > p.
%F A378220 a(n) = A023900(A003961(n)).
%F A378220 For n > 1, a(n) = 2*A349385(n).
%F A378220 a(n) = Sum_{d|n} A346234(d).
%F A378220 a(n) = Sum_{d|n} A346246(d)*A378216(n/d).
%t A378220 f[p_, e_] := 1 - NextPrime[p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jan 13 2025 *)
%o A378220 (PARI) A378220(n) = factorback(apply(p -> 1-nextprime(1+p), factor(n)[, 1]));
%o A378220 (Python)
%o A378220 from math import prod
%o A378220 from sympy import nextprime, primefactors
%o A378220 def A378220(n): return prod(1-nextprime(p) for p in primefactors(n)) # _Chai Wah Wu_, Nov 23 2024
%Y A378220 Cf. A023900, A003961, A048673, A151800, A349385, A378216.
%Y A378220 Dirichlet inverse of A003972.
%Y A378220 Inverse Möbius transform of A346234.
%Y A378220 After the initial term, A349385 doubled.
%Y A378220 Cf. A346246, A378216.
%K A378220 sign,mult
%O A378220 1,2
%A A378220 _Antti Karttunen_, Nov 23 2024