A143189 Numbers k such that at least one of the following equations holds: phi(2*k-1) = phi(2*k), sigma(2*k-1) = sigma(2*k), tau(2*k-1) = tau(2*k).

1, 2, 8, 11, 17, 29, 38, 43, 47, 67, 68, 71, 73, 74, 86, 89, 95, 101, 103, 107, 109, 116, 122, 127, 128, 143, 151, 188, 191, 194, 197, 215, 223, 227, 241, 248, 251, 254, 269, 277, 283, 302, 305, 317, 323, 332, 349, 359, 368, 371, 373, 397, 403, 409, 410, 433, 452

Offset: 1

Author

Roger L. Bagula, Oct 17 2008

Keywords

• nonn

Comments

Inspired by A134922.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005 (tau), A000010 (phi), A000203 (sigma), A134922.

Programs

Mathematica

a[n_] =
If[EulerPhi[2*n - 1] - EulerPhi[2*n] == 0, n,
If[DivisorSigma[1, 2*n - 1] - DivisorSigma[1, 2*n] == 0, n,
If[Length[Divisors[2*n - 1]] - Length[Divisors[2*n]] == 0, n, {}]]];
Flatten[Table[a[n], {n, 1, 400}]]

PARI

isok(k) = {my(f1 = factor(2*k-1), f2 = factor(2*k)); eulerphi(f1) == eulerphi(f2) || sigma(f1) == sigma(f2) || numdiv(f1) == numdiv(f2);} \\ Amiram Eldar, Jun 07 2025

Extensions

Name edited by Amiram Eldar, Jun 07 2025