A186749 a(n) = phi(n - phi(n) + 3).

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

Offset: 1

Author

Wesley Ivan Hurt, Aug 29 2013

Keywords

• nonn

Comments

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

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.

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.

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537

Crossrefs

Cf. A000010, A051953.

Programs

GAP

List([1..70],n->Phi(n-Phi(n)+3)); # Muniru A Asiru, Mar 04 2018

Magma

[EulerPhi(n-EulerPhi(n)+3): n in [1..100]]; // Vincenzo Librandi, Dec 08 2015

Maple

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

Mathematica

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

PARI

A186749(n) = eulerphi(n - eulerphi(n) + 3); \\ Antti Karttunen, Mar 04 2018

Formula

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