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

Author

T. D. Noe, Aug 21 2008

Keywords

• more
• nonn

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

• 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

Crossrefs

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

Programs

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

Extensions

Definition corrected by Max Alekseyev, Oct 14 2023