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A083876 Least pseudoprime to base 2 through base prime(n).

%I A083876 #71 Dec 18 2022 11:46:15
%S A083876 341,1105,1729,29341,29341,162401,252601,252601,252601,252601,252601,
%T A083876 252601,1152271,2508013,2508013,3828001,3828001,3828001,3828001,
%U A083876 3828001,3828001,3828001,3828001,3828001,3828001,6733693,6733693,6733693
%N A083876 Least pseudoprime to base 2 through base prime(n).
%C A083876 Records: 341, 1105, 1729, 29341, 162401, 252601, 1152271, 2508013, 3828001, 6733693, 17098369, 17236801, 29111881, 82929001, 172947529, 216821881, 228842209, 366652201, .... - _Robert G. Wilson v_, May 11 2012
%C A083876 Conjecture: for n > 1, a(n) is the smallest Carmichael number k with lpf(k) > prime(n). It seems that such Carmichael numbers have exactly three prime factors. - _Thomas Ordowski_, Apr 18 2017
%C A083876 The conjecture is true if a(n) < A285549(n) for all n > 1. It holds for all a(n) < 2^64. - _Max Alekseyev_ and _Thomas Ordowski_, Mar 13 2018
%C A083876 If prime(n) < m < a(n), then m is prime if and only if p^(m-1) == 1 (mod m) for every prime p <= prime(n). - _Thomas Ordowski_, Mar 05 2018
%C A083876 By this conjecture in the second comment, a(n) <= A135720(n+1), with equality for n > 1 iff a(n) < a(n+1), namely for n = 2, 3, 5, 6, 12, 13, 15, 25, 28, 29, ... For such n, a(n) gives all terms of A300629 > 561. - _Thomas Ordowski_, Mar 10 2018
%H A083876 Giovanni Resta, <a href="/A083876/b083876.txt">Table of n, a(n) for n = 1..1000</a> (first 100 terms from Robert G. Wilson v)
%t A083876 k = 4; Do[l = Table[ Prime[i], {i, 1, n}]; While[ PrimeQ[k] || Union[PowerMod[l, k - 1, k]] != {1}, k++ ]; Print[k], {n, 1, 29}]
%o A083876 (PARI) isps(k, n) = {if (isprime(k), return (0)); my(nbok = 0); for (b=2, prime(n), if (Mod(b, k)^(k-1) == 1, nbok++, break)); if (nbok==prime(n)-1, return (1));}
%o A083876 a(n) = {my(k=2); while (!isps(k, n), k++); return (k);} \\ _Michel Marcus_, Apr 27 2018
%Y A083876 Cf. A002997, A001567, A052155, A083737, A087788, A083739, A135720, A141768, A250199, A271221, A285549, A300629.
%K A083876 nonn
%O A083876 1,1
%A A083876 _Robert G. Wilson v_, May 06 2003