

A293928


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


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
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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 ab implies phi(a)phi(b), and for k >= 1 a^(k1)a^k.
Where (if ever) does this first differ from A068997?  R. J. Mathar, Oct 30 2017


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: A320580 A325763 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



