A143510 Numbers m such that the equation phi(x) = m has even but no odd solutions.

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

Table of n, a(n) for n=1..9.

Max Alekseyev, PARI scripts for various problems

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

Cf. A143511 (least k such that phi(k)=m).

Sequence in context: A230636 A283029 A250933 * A043680 A204673 A205640

Adjacent sequences: A143507 A143508 A143509 * A143511 A143512 A143513

Keyword

more,nonn

Author

T. D. Noe, Aug 21 2008

Extensions

Definition corrected by Max Alekseyev, Oct 14 2023

Status

approved