OFFSET
1,1
COMMENTS
In the unlikely event that Carmichael's conjecture is proved false, the counterexamples will be in this sequence. The number a(1) = 16842752 = 257*2^16 is mentioned in problem E3361. If there are only five Fermat primes, then 2^k is in this sequence for all k>31. It appears that for every product d of Fermat primes (A143512), the number 2^k * d is in this sequence for some k. The link to "Numbers Like 16842752" lists examples for various d.
Conjecture: if the least solution to phi(x) = m is even, then m is in this sequence. - Jianing Song, Nov 07 2022
The conjecture holds for the first 26 such numbers, see A387221. - Jud McCranie, Dec 17 2025
REFERENCES
R. K. Guy, Unsolved problems in number theory, B39.
LINKS
Jud McCranie, Table of n, a(n) for n = 1..257
Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).
T. D. Noe, Numbers Like 16842752
William P. Wardlaw, L. L. Foster and R. J. Simpson, Problem E3361, Amer. Math. Monthly, Vol. 98, No. 5 (May, 1991), 443-444.
Eric Weisstein's World of Mathematics, Carmichael's Totient Function Conjecture
PROG
(PARI) isok(k) = numinvphi(k) && select(x->((x%2) == 1), invphi(k)) == 0; \\ using invphi from PARI scripts link; Michel Marcus, Oct 09 2023; corrected by Max Alekseyev, Oct 14 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
T. D. Noe, Aug 21 2008
EXTENSIONS
Definition corrected by Max Alekseyev, Oct 14 2023
STATUS
approved