[seqfan] A051903 and generalization of Lehmer's totient problem

[Tomasz Ordowski]

Hello SeqFans!

Let a(n) = A051903(n) : https://oeis.org/A051903
a(n) is the smallest k such that b^{phi(n)+k} == b^k (mod n) for all b.
The Euler phi function can be replaced by the Carmichael lambda function.

The Problem (Amiram Eldar did not find any solution n < 10^8):
Are there composite numbers n > 4 such that n == a(n) (mod phi(n))?
By Lehmer's totient conjecture, there are no such squarefree numbers.

Let's define:
Numbers n such that a(n) > 1 and n == a(n) (mod lambda(n)).
4, 12, 16, 48, 80, 112, 132, 208, 240, 1104, 1456, 1892, 2128, 4144, 5852,
12208, 17292, 18544, 21424, 25456, 30160, 45904, 78736, 97552, 106384,
138864, 153596, 154960, 160528, 289772, 311920, 321904, 399212, 430652,
545584, 750064, 770704, 979916, 1037040, 1058512, 1118416, 1193852,
1229584, 1296880, 1474672, 1687504, ... [data from Amiram Eldar].
Are there infinitely many such numbers?
Are all such numbers even?

Best regards,

Thomas Ordowski
_______________________
Cf. https://oeis.org/A276976 (see the last formula).
http://mathworld.wolfram.com/KnoedelNumbers.html
http://mathworld.wolfram.com/LehmersTotientProblem.html