A293928 Totients phi(m) having one or more solutions to phi(m)^(k+1) = phi(phi(m)^k*m), k >= 1, m >= 1.

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

1,2

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 of phi(m)^(k+1) = phi(phi(m)^k*m), k >= 1, m >= 1.

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), ...

If a solution exists then the smallest value of k must be 1. This follows from a|b implies phi(a)|phi(b), and for k >= 1 a^(k-1)|a^k.

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

Links

Table of n, a(n) for n=1..59.

Max Alekseyev, PARI scripts for various problems

Formula

0 < phi(m)^(k+1) = phi(phi(m)^k*m), k >= 1, m >= 1.

Example

96 is a term since 96^2 = phi(96*288), with k=1 and m=288 where phi(288) = 96.

Prog

(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

Crossrefs

Cf. A000010, A006511, A032447, A007366.

Subsequence of A002202.

Sequence in context: A325763 A363949 A068997 * A067712 A060765 A140110

Adjacent sequences: A293925 A293926 A293927 * A293929 A293930 A293931

Keyword

nonn

Author

Torlach Rush, Oct 19 2017

Extensions

More terms from Michel Marcus, Oct 24 2017

Status

approved