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A373580
Numbers k that divide 2^(2^k) - 2^k + 1.
2
1, 3, 5, 17, 257, 641, 1605, 4369, 32113, 65537, 94945, 114689, 159441, 164737, 274177, 319489, 974849, 2424833, 3862465, 6700417, 13631489, 13906833, 16843009, 26017793, 42009217, 45592577, 63766529, 70463489, 167772161, 175747457, 825753601, 1214251009, 1227890731, 1251711641
OFFSET
1,2
COMMENTS
Numbers k that divide A119564(k).
A term is prime if and only if it is in A023394.
If k is in A307843, then it is a term of this sequence.
The terms that are not divisors of Fermat numbers are 1605, 4369, 32113, 94945, ... (they are all composite). Are there infinitely many of them?
Note that 2^(2^k) - 2^k + 1 = (2^(2^k) - 1) - (2^k - 2).
MATHEMATICA
q[k_] := Mod[PowerMod[2, 2^k, k] - PowerMod[2, k, k] + 1, k] == 0; Select[Range[1, 10^5, 2], q] (* Amiram Eldar, Jun 10 2024 *)
PROG
(PARI) isok(k) = Mod(Mod(2, k)^(2^k) - Mod(2, k)^k + 1, k) == 0; \\ Michel Marcus, Jun 12 2024
CROSSREFS
Cf. A023394 (primes in this sequence), A119564, A307843 (subsequence).
Sequence in context: A272060 A333873 A058910 * A307843 A023394 A176689
KEYWORD
nonn
AUTHOR
Thomas Ordowski, Jun 10 2024
EXTENSIONS
More terms from Amiram Eldar, Jun 10 2024
STATUS
approved