A202855 Numbers n such that phi(n) - 1 divides n, where phi is Euler's totient function.

3, 4, 6, 12, 60, 1020, 262140, 334495740, 17179869180, 27971850688528380

Offset: 1

Author

José María Grau Ribas, Dec 25 2011

Keywords

• nonn
• more

Comments

The sequence b(n) = 4*A050474(n) is a subsequence of this sequence, and comprises solutions of n/(phi(n) - 1) = 4, accounting for all terms up to a(9) except a(1) and a(3). Proof: suppose n/(phi(n) - 1) = 4. With n = 4*x, x/(phi(4*x) - 1) = 1, or phi(4*x) = x + 1. Since phi(k) is even for k > 2, x is odd, and phi(4*x) = 2*phi(x) = x + 1, the definition of A050474. It follows that 4*A050474(8) = 27971850688528380 is a term of this sequence. - Chris Boyd, Mar 22 2015

Similarly, the terms with n/(phi(n) - 1) = 3 are given by 3 * terms of A050474 coprime to 3; n/(phi(n) - 1) = 6 are given by 6 * terms of A050474 coprime to 6. Also, the terms of n/(phi(n) - 1) = 5 are given by 5 * terms t of A203966 coprime to 5 and having (t+1)/phi(t) = 4. Note that n/(phi(n) - 1) = 2 is impossible. - Max Alekseyev, Oct 26 2023

Crossrefs

Cf. A000010, A050474, A203966.

Programs

Mathematica

Select[1 + Range[1000000], Divisible[#, EulerPhi[#] - 1] &]

PARI

for(n=3,1e7,if(n%(eulerphi(n)-1)==0,print1(n", "))) \\ Charles R Greathouse IV, Dec 26 2011

Extensions

a(8) from Charles R Greathouse IV, Dec 27 2011

a(9) from Donovan Johnson, Dec 29 2011

a(10) from Chris Boyd confirmed by Max Alekseyev, Oct 26 2023