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A048950 Base-3 Euler-Jacobi pseudoprimes.

%I A048950 #32 Feb 16 2025 08:32:40
%S A048950 121,703,1729,1891,2821,3281,7381,8401,8911,10585,12403,15457,15841,
%T A048950 16531,18721,19345,23521,24661,28009,29341,31621,41041,44287,46657,
%U A048950 47197,49141,50881,52633,55969,63139,63973,74593,75361,79003,82513
%N A048950 Base-3 Euler-Jacobi pseudoprimes.
%C A048950 Odd composite k with gcd(k,3) = 1 and 3^((k-1)/2) == (3,k) (mod k) where (.,.) is the Jacobi symbol. - _R. J. Mathar_, Jul 15 2012
%C A048950 The base 5 Euler-Jacobi pseudoprimes are 781, 1541, 1729, 5461, 5611, 6601, 7449, ... - _R. J. Mathar_, Jul 15 2012 [Typo fixed; this is A375914. - _Jianing Song_, Sep 02 2024]
%H A048950 Amiram Eldar, <a href="/A048950/b048950.txt">Table of n, a(n) for n = 1..10000</a>
%H A048950 A. Rotkiewicz, <a href="https://doi.org/10.1090/S0025-5718-1982-0658229-0">On Euler Lehmer pseudoprimes and strong Lehmer pseudoprimes with Parameters L, Q in arithmetic progressions</a>, Math. Comp 39 (159) (1982) 239-247.
%H A048950 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Euler-JacobiPseudoprime.html">Euler-Jacobi Pseudoprime</a>.
%H A048950 <a href="/index/Ps#pseudoprimes">Index entries for sequences related to pseudoprimes</a>
%t A048950 Select[Range[1, 10^5, 2], GCD[#, 3] == 1 && CompositeQ[#] && PowerMod[3, (# - 1)/2, #] == Mod[JacobiSymbol[3, #], #] &] (* _Amiram Eldar_, Jun 28 2019 *)
%o A048950 (PARI) is(n) = n%2==1 && gcd(n,3)==1 && Mod(3, n)^((n-1)/2)==kronecker(3,n)
%o A048950 forcomposite(c=1, 83000, if(is(c), print1(c, ", "))) \\ _Felix Fröhlich_, Jul 15 2019
%Y A048950 Cf. A005935.
%Y A048950                                    |        b=2        |   b=3    |   b=5   |
%Y A048950 -----------------------------------+-------------------+----------+---------+
%Y A048950   (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917  | A375915 |
%Y A048950 -----------------------------------+-------------------+----------+---------+
%Y A048950  (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918  | A375916 |
%Y A048950 -----------------------------------+-------------------+----------+---------+
%Y A048950  b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490  | A375816 |
%Y A048950  (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |          |         |
%Y A048950 -----------------------------------+-------------------+----------+---------+
%Y A048950      Euler-Jacobi pseudoprimes     |      A047713      | this seq | A375914 |
%Y A048950        (union of first two)        |                   |          |         |
%Y A048950 -----------------------------------+-------------------+----------+---------+
%Y A048950         Euler pseudoprimes         |      A006970      | A262051  | A262052 |
%Y A048950        (union of all three)        |                   |          |         |
%K A048950 nonn
%O A048950 1,1
%A A048950 _Eric W. Weisstein_