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A380354 a(n) = phi(2 + phi(3 + phi(5 + ... + phi(prime(n))))), where phi is Euler totient function (A000010).

%I A380354 #20 Jan 27 2025 06:51:53
%S A380354 1,2,4,6,4,8,8,12,12,16,20,20,18,40,40,16,18,18,16,72,40,16,40,18,96,
%T A380354 96,18,64,20,40,20,48,42,40,42,20,20,40,40,20,18,20,64,64,20,40,40,40,
%U A380354 40,20,40,20,18,64,64,40,40,20,40,20,40,64,20,40,40,20,20,64,64,64
%N A380354 a(n) = phi(2 + phi(3 + phi(5 + ... + phi(prime(n))))), where phi is Euler totient function (A000010).
%C A380354 Inspired by A380340, A380341 and A380342.
%C A380354 Conjecture 1: a(n) can be only 1, 2, 4, 6, 8, 12, 16, 20, 18, 40, 72, 96, 64, 48 or 42.
%C A380354 Conjecture 2: for n >= 187, a(n) can be only 20 or 64.
%H A380354 Paolo Xausa, <a href="/A380354/b380354.txt">Table of n, a(n) for n = 1..1000</a>
%t A380354 A380354[n_] := Fold[EulerPhi[#2 + #] &, 0, Prime[Range[n, 1, -1]]];
%t A380354 Array[A380354, 100]
%o A380354 (PARI) a(n) = my(x=0); forstep(k=n, 1, -1, x = eulerphi(prime(k)+x)); x; \\ _Michel Marcus_, Jan 22 2025
%o A380354 (Python)
%o A380354 from functools import reduce
%o A380354 from sympy import totient, primerange
%o A380354 def A380354(n): return totient(reduce(lambda x,y:totient(x)+y,tuple(reversed(tuple(primerange(prime(n)+1)))))) # _Chai Wah Wu_, Jan 23 2025
%Y A380354 Cf. A000010, A380340, A380341, A380342, A380414, A380415.
%K A380354 nonn
%O A380354 1,2
%A A380354 _Paolo Xausa_, Jan 22 2025