A293928 Totients phi(m) having one or more solutions m to phi(m)^2 = phi(phi(m)*m).

1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 32, 36, 40, 48, 54, 64, 72, 80, 84, 96, 100, 108, 120, 128, 144, 160, 162, 168, 192, 200, 216, 240, 252, 256, 272, 288, 312, 320, 324, 336, 360, 384, 400, 432, 440, 480, 486, 500, 504, 512, 544, 576, 588, 600, 624, 640, 648, 672, 684

Offset: 1

Author

Torlach Rush, Oct 19 2017

Keywords

• nonn

Comments

"Totients" are terms of A000010. - N. J. A. Sloane, Oct 22 2017

The smallest totient absent from the list is 10. This is because the totient inverses of 10, 11 and 22 are not solutions to phi(m)^2 = phi(phi(m)*m).

The formula is recursive. For example, taking a(22) we get the following: 11664 = phi(108*324), 1259712 = phi(11664*324), 136048896 = phi(1259712*324), ...

Where (if ever) does this first differ from A068997? - R. J. Mathar, Oct 30 2017

Apparently the set of the m is A151999. - R. J. Mathar, Mar 25 2024

If m satisfies phi(m)^2 = phi(phi(m)*m), then it satisfies phi(m)^(k+1) = phi(phi(m)^k*m) for all k >= 1. - Max Alekseyev, Dec 03 2024

Examples

			96 is a term since 96^2 = phi(96*288), with m=288 where phi(288) = 96.
		

Links

• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Crossrefs

Cf. A000010, A006511, A032447, A007366.

Subsequence of A002202.

Programs

PARI

isok(n) = {my(iv = invphi(n)); if (#iv, for (m = 1, #iv, if (n^2 == eulerphi(n*iv[m]), return (1)););); return (0);} \\ using the invphi script by Max Alekseyev; Michel Marcus, Nov 01 2017

Extensions

More terms from Michel Marcus, Oct 24 2017

Definition simplified by Max Alekseyev, Dec 03 2024