A186749 - OEIS

%I #24 Sep 08 2022 08:45:56

%S 2,2,2,4,2,6,2,6,2,6,2,10,2,10,4,10,2,8,2,8,4,8,2,18,4,16,4,18,2,20,2,

%T 18,8,12,6,18,2,22,6,18,2,20,2,18,8,18,2,24,4,20,10,30,2,24,6,24,8,20,

%U 2,46,2,24,8,24,8,42,2,24,12,42,2,32,2,40,18,42

%N a(n) = phi(n - phi(n) + 3).

%C This is the Euler Phi function of 3 more than the Cototient of n.

%C If n is noncomposite, a(n) = 2. Proof: For n = 1, phi(1 - phi(1) + 3) = phi(1-1+3) = phi(3) = 2. For n = p, phi(p - phi(p) + 3) = phi(p - (p-1) + 3) = phi(4) = 2.

%C If n is the product of twin primes, a(n) is the arithmetic mean of the prime factors. Equivalently, when n is the product of twin primes, a(n) +- 1 represents the largest and the smallest prime factors of n respectively.

%H Antti Karttunen, <a href="/A186749/b186749.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = phi(n - phi(n) + 3) = A000010(n - A000010(n) + 3) = A000010(A051953(n) + 3).

%e a(15) = 4, Since phi(15 - phi(15) + 3) = 4. Note that 15 is the product of twin primes and that a(15) = 4 is the arithmetic mean of the prime factors of 15: (3+5)/2 = 4.

%p with(numtheory); seq( phi(k - phi(k) + 3), k=1..70);

%t Table[EulerPhi[n - EulerPhi[n] + 3], {n, 100}]

%o (Magma) [EulerPhi(n-EulerPhi(n)+3): n in [1..100]]; // _Vincenzo Librandi_, Dec 08 2015

%o (PARI) A186749(n) = eulerphi(n - eulerphi(n) + 3); \\ _Antti Karttunen_, Mar 04 2018

%o (GAP) List([1..70],n->Phi(n-Phi(n)+3)); # _Muniru A Asiru_, Mar 04 2018

%Y Cf. A000010, A051953.

%K nonn

%O 1,1

%A _Wesley Ivan Hurt_, Aug 29 2013