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%I #30 Apr 29 2019 11:00:54
%S 3,5,13,17,41,67,197,257,523,1031,2069,4129,8243,16451,32911,65537,
%T 131479,262153,524591,1048601,2097229,4194389,8388791,16777441,
%U 33554501,67108913,134217757,268435889,536871259,1073741971,2147484949,4294967297,8589934651,17179869827
%N a(n) is the smallest k > 2^n such that 2^(k-1) == 1 (mod (2^n-1)*k).
%C a(n) = smallest k > 2^n such that k == 1 (mod n) and 2^(k-1) == 1 (mod k), so a(n) is an odd prime or a pseudoprime (Fermat pseudoprime to base 2).
%C Conjecture: a(n) is composite if and only if n = 2^j and 2^(2^j) + 1 is composite (presumably for all j > 4).
%C Note that a(2^j) = 2^(2^j) + 1 = A000215(j), the Fermat numbers.
%C For n <> 2^j, a(n) is the smallest k = 2^n - (2^n mod n) + m*n + 1 for m > 0 such that 2^(k-1) == 1 (mod k).
%C The last definition, also without the condition n <> 2^j, probably gives only primes.
%F a(n) == 1 (mod n).
%F a(2^j) = A000215(j).
%e a(32) = 2^(2^5) + 1 = 641*6700417 is the smallest composite term.
%t a[n_] := Module[{k = 2^n + 1}, While[PowerMod[2, k - 1, (2^n - 1)*k] != 1, k++]; k]; Array[a, 50] (* _Amiram Eldar_, Apr 12 2019 *)
%o (PARI) a(n) = my(k=2^n+1); while( Mod(2, (2^n-1)*k)^(k-1) != 1, k++); k; \\ _Michel Marcus_, Apr 25 2019
%Y Cf. A000215, A000225, A001567, A065091, A270427, A306826.
%K nonn
%O 1,1
%A _Thomas Ordowski_, Apr 12 2019
%E More terms from _Amiram Eldar_, Apr 12 2019
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