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A143510
Numbers m such that the equation phi(x) = m has even but no odd solutions.
3
16842752, 33685504, 67371008, 134742016, 269484032, 538968064, 1077936128, 2155872256, 4294967296
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
REFERENCES
R. K. Guy, Unsolved problems in number theory, B39.
LINKS
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
Cf. A143511 (least k such that phi(k)=m).
Sequence in context: A230636 A283029 A250933 * A043680 A204673 A205640
KEYWORD
more,nonn,changed
AUTHOR
T. D. Noe, Aug 21 2008
EXTENSIONS
Definition corrected by Max Alekseyev, Oct 14 2023
STATUS
approved