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A329468
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Carmichael numbers all of whose prime factors are congruent to 3 modulo 4.
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2
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8911, 1024651, 1152271, 1773289, 5481451, 8830801, 9585541, 10267951, 14913991, 15888313, 26474581, 40917241, 45877861, 64377991, 67902031, 72108421, 72286501, 81926461, 94536001, 104852881, 111291181, 129762001, 139592101, 139952671, 178482151, 213835861, 368113411
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OFFSET
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1,1
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COMMENTS
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Galbraith et al. (2019) proved that for a Carmichael number m, the number of bases below m in which m is a strong pseudoprime is S(m) = A071294((m-1)/2) <= phi(m)/2^(omega(m)-1), with equality when m is a term of this sequence, where phi is the Euler totient function (A000010) and omega(m) is the number of distinct prime factors of m (A001221).
The corresponding values of S(a(n)) are 1782, 240570, 277830, 176418, 1316250, 882090, 984150, 2515590, 3611790, 1587762, ...
The least term with 3, 4, 5, ... prime factors is 8911, 1773289, 1419339691, 4077957961, 3475350807391, 440515336876021, 574539328092938671, 2426698123549677901, ...
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LINKS
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Steven Galbraith, Jake Massimo and Kenneth G. Paterson, Safety in Numbers: On the Need for Robust Diffie-Hellman Parameter Validation, in: Dongdai Lin and Kazue Sako (eds.), Public-Key Cryptography - PKC 2019, 22nd IACR International Conference on Practice and Theory of Public-Key Cryptography, Beijing, China, April 14-17, 2019, Proceedings, Part II, Springer, 2019.
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EXAMPLE
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8911 = 7 * 19 * 67 is a term since it is a Carmichael number, and 7 == 19 == 67 == 3 (mod 4).
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MATHEMATICA
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aQ[n_] := CompositeQ[n] && Divisible[n - 1, CarmichaelLambda[n]] && AllTrue[ FactorInteger[n][[;; , 1]], Mod[#, 4] == 3 &]; Select[Range[2*10^6], aQ]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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