cp's OEIS Frontend

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

%I A143510 #24 Feb 16 2025 08:33:08
%S A143510 16842752,33685504,67371008,134742016,269484032,538968064,1077936128,
%T A143510 2155872256,4294967296
%N A143510 Numbers m such that the equation phi(x) = m has even but no odd solutions.
%C A143510 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.
%C A143510 Conjecture: if the least solution to phi(x) = m is even, then m is in this sequence. - _Jianing Song_, Nov 07 2022
%D A143510 R. K. Guy, Unsolved problems in number theory, B39.
%H A143510 Max Alekseyev, <a href="https://oeis.org/wiki/User:Max_Alekseyev/gpscripts">PARI/GP Scripts for Miscellaneous Math Problems</a> (invphi.gp).
%H A143510 T. D. Noe, <a href="http://www.sspectra.com/math/16842752.txt">Numbers Like 16842752</a>
%H A143510 William P. Wardlaw, L. L. Foster and R. J. Simpson, <a href="http://www.jstor.org/stable/2323869">Problem E3361</a>, Amer. Math. Monthly, Vol. 98, No. 5 (May, 1991), 443-444.
%H A143510 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CarmichaelsTotientFunctionConjecture.html">Carmichael's Totient Function Conjecture</a>
%o A143510 (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
%Y A143510 Cf. A143511 (least k such that phi(k)=m).
%K A143510 more,nonn
%O A143510 1,1
%A A143510 _T. D. Noe_, Aug 21 2008
%E A143510 Definition corrected by _Max Alekseyev_, Oct 14 2023