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A262084
Numbers n that satisfy the equation phi(n + 6) = phi(n) + 6 where phi(n) = A000010(n) is Euler's totient function.
4
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.
A natural number n is a solution if either:
n and n + 6 are both prime (sexy primes) (A023201);
n = 2^k*p with k>0 and p prime, such that 2^k*(p+1) - 1 is also prime.
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. A001838 (k=2), A056772 (k=4), A262085 (k=8), A262086 (k=10).
Sequence in context: A024891 A277907 A136144 * A086304 A193947 A120222
KEYWORD
nonn,easy
AUTHOR
Kevin J. Gomez, Sep 10 2015
STATUS
approved