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a(n) is the smallest base a > 2 such that a^(k-1) != 1 (mod k), where k = A001567(n), the n-th Fermat pseudoprime to base 2.
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%I #24 Nov 27 2018 03:55:14

%S 3,3,3,5,3,7,3,3,5,5,7,3,3,3,3,3,3,7,3,3,3,7,3,5,3,3,3,3,3,3,3,7,3,3,

%T 5,3,3,3,3,13,3,3,3,3,5,3,3,3,3,7,3,3,13,5,3,7,3,3,3,3,3,7,3,3,3,3,3,

%U 11,3,5,5,3,3,3,5,5,3,5,7,5,5,3,13,3,3

%N a(n) is the smallest base a > 2 such that a^(k-1) != 1 (mod k), where k = A001567(n), the n-th Fermat pseudoprime to base 2.

%C a(n) <= A177415(n).

%C Each a(n) is an odd prime.

%C If k = A001567(n) is a Carmichael number, then a(n) = lpf(k).

%C Conjecture: if k = A001567(n) is semiprime, then a(n) < lpf(k).

%C The smallest numbers k = A001567(n) such that a(n) = prime(m) for m > 1 are 341, 1105, 1729, 75361, 29341, 162401, 334153, ... See A135720 > 561.

%C The smallest such semiprimes are 341, 2701, ?, 721801, ... Cf. A285549.

%e The first Fermat pseudoprime to base 2 is 341, and 341 is not a Fermat pseudoprime to base 3, so a(1) = 3.

%t a[p_] := Module[{m=3}, While[Mod[m^(p-1), p] == 1, m++]; m]; psp = Select[Range[3, 1000000, 2], CompositeQ[ # ] && PowerMod[2, (# - 1), # ] == 1 &]; Map[a, psp] (* _Amiram Eldar_, Nov 19 2018 *)

%Y Cf. A001567, A002997, A083876, A135720, A177415, A214305, A285549.

%Y A141710 is a subsequence.

%K nonn

%O 1,1

%A _Thomas Ordowski_, Nov 19 2018

%E More terms from _Amiram Eldar_, Nov 19 2018