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A171262

Numbers k such that phi(k) = 2*phi(k+1).

5, 13, 35, 37, 61, 73, 157, 193, 277, 313, 397, 421, 455, 457, 541, 613, 661, 665, 673, 733, 757, 877, 997, 1085, 1093, 1153, 1201, 1213, 1237, 1295, 1321, 1381, 1453, 1621, 1657, 1753, 1873, 1933, 1993, 2017, 2137, 2169, 2341, 2473, 2557, 2593, 2797, 2857

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OFFSET

1,1

COMMENTS

Theorem: A prime p is in the sequence iff (1/2)*(p+1) is prime.

Proof: If both numbers p and 1/2*(p+1) are prime then phi(p) = p-1=2*(p-1)/2 and 2*(1/2*(p+1)-1) = 2*phi(1/2*(p+1)), (1/2)*(p+1) is odd so phi(1/2*(p+1)) = phi(p+1) hence phi(p) = 2*phi(p+1), namely p is in the sequence.

Also if p is a prime term of the sequence then phi(p) = 2*phi(p+1) so p-1 = 2*phi(p+1) or phi(p+1) = (1/2)*(p+1)-1 hence (1/2)*(p+1)is prime.

LINKS

Ray Chandler, Table of n, a(n) for n = 1..10000

EXAMPLE

phi(35) = 2*12 = 2*phi(35+1), so 35 is in the sequence.

MATHEMATICA

Select[Range[2900], EulerPhi[ # ]==2EulerPhi[ #+1]&]

PROG

(Magma) [n: n in [1..3*10^3] | EulerPhi(n) eq 2*EulerPhi(n+1)]; // Vincenzo Librandi, Apr 14 2015

CROSSREFS

Cf. A000010 (phi), A005383, A171271.

Sequence in context: A034521 A294841 A092647 * A006561 A146845 A192310