A262084 Numbers m that satisfy the equation phi(m + 6) = phi(m) + 6 where phi(m) = A000010(m) is Euler's totient function.

5, 7, 11, 13, 17, 21, 23, 31, 37, 40, 41, 47, 53, 56, 61, 67, 73, 83, 88, 97, 98, 101, 103, 107, 131, 136, 151, 152, 156, 157, 167, 173, 191, 193, 223, 227, 233, 237, 248, 251, 257, 263, 271, 277, 296, 307, 311, 328, 331, 347, 353, 367, 373, 376, 383, 433, 443

Offset: 1

Author

Kevin J. Gomez, Sep 10 2015

Keywords

• nonn
• easy

Comments

The majority of solutions can be predicted by known properties of the equality. There are several solutions that do not fit these parameters.

An odd natural number m is a solution if m and m + 6 are both prime (sexy primes) (A023201).

Among the solutions for even natural numbers are all m = 8*p with odd primes p such that 4*p+3 is a prime number. Proof: From A000010 we can learn that the formula phi(p*2) = floor(((2 + p - 1) mod p)/(p - 1)) + p - 1 is known. If we define p = 4*q+3 and m = 8*q and insert, we will obtain phi(8*q+6) = 4*q+2. Also it is known that phi(8*q) = 4*q-4 if q is any odd prime. - Thomas Scheuerle, Dec 20 2024

Examples

			5 is a term since phi(5+6) = 10 = 6 + 4 = phi(5) + 6.
		

Links

• Peter Kagey, Table of n, a(n) for n = 1..10000
• Wikipedia, Euler's totient function

Crossrefs

Cf. A000010, A023201.

Cf. A001838 (k=2), A056772 (k=4), A262085 (k=8), A262086 (k=10).

Programs

Magma

[n: n in [1..500] | EulerPhi(n+6) eq EulerPhi(n)+6]; // Vincenzo Librandi, Sep 11 2015

Mathematica

Select[Range@500, EulerPhi@(# + 6)== EulerPhi[#] + 6 &] (* Vincenzo Librandi, Sep 11 2015 *)

PARI

is(n)=eulerphi(n + 6) == eulerphi(n) + 6 \\ Anders Hellström, Sep 11 2015

Sage

[n for n in [1..1000] if euler_phi(n+6)==euler_phi(n)+6] # Tom Edgar, Sep 10 2015