OFFSET
1,1
COMMENTS
Conjecture: for k > 1, (2^k-1)*k^2 divides (k-1)^(2^k-2)-1 if and only if 2^k-1 is (a Mersenne) prime or k = 2^(2^m)+1 (is a Fermat number).
If so, composite terms are Fermat composite numbers k = 2^(2^m)+1. The smallest (for m = 5) is k = 4294967297 = 641*6700417 and probably all larger (for m > 5).
If my conjecture is true, primes in this sequence that are not Mersenne exponents are Fermat prime numbers k = 2^(2^m)+1 for m > 2, namely k = 257, 65537 and there are probably no more.
MATHEMATICA
q[n_] := PowerMod[n - 1, 2^n - 2, n^2*(2^n - 1)] == 1; Select[Range[1000], q] (* Amiram Eldar, Oct 02 2023 *)
PROG
(Python)
def A366029_gen(startvalue=1): # generator of terms>= startvalue
return filter(lambda k:pow(k-1, (m:=(1<<k)-1)-1, m*k**2)==1, count(max(startvalue, 1)))
CROSSREFS
KEYWORD
nonn,more,hard
AUTHOR
Thomas Ordowski, Oct 01 2023
EXTENSIONS
More terms from Amiram Eldar, Oct 02 2023
STATUS
approved