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

1,1

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

Links

Peter Kagey, Table of n, a(n) for n = 1..10000

Wikipedia, Euler's totient function

Example

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

Mathematica

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

Prog

(Sage) [n for n in [1..1000] if euler_phi(n+6)==euler_phi(n)+6] # Tom Edgar, Sep 10 2015
(Magma) [n: n in [1..500] | EulerPhi(n+6) eq EulerPhi(n)+6]; // Vincenzo Librandi, Sep 11 2015
(PARI) is(n)=eulerphi(n + 6) == eulerphi(n) + 6 \\ Anders Hellström, Sep 11 2015

Crossrefs

Cf. A000010, A023201.

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

Sequence in context: A024891 A277907 A136144 * A086304 A193947 A120222

Adjacent sequences: A262081 A262082 A262083 * A262085 A262086 A262087

Keyword

nonn,easy

Author

Kevin J. Gomez, Sep 10 2015

Status

approved