cp's OEIS Frontend

A375915 Composite numbers k == 1, 9 (mod 10) such that 5^((k-1)/2) == 1 (mod k).

%I A375915 #20 Sep 07 2024 16:03:49
%S A375915 781,1541,1729,5461,5611,6601,7449,11041,12801,13021,14981,15751,
%T A375915 15841,21361,24211,25351,29539,38081,40501,41041,44801,47641,53971,
%U A375915 67921,75361,79381,90241,100651,102311,104721,106201,106561,112141,113201,115921,133141,135201,141361
%N A375915 Composite numbers k == 1, 9 (mod 10) such that 5^((k-1)/2) == 1 (mod k).
%C A375915 Odd composite numbers k such that 5^((k-1)/2) == (5/k) = 1 (mod k), where (5/k) is the Jacobi symbol (or Kronecker symbol).
%H A375915 Jianing Song, <a href="/A375915/b375915.txt">Table of n, a(n) for n = 1..1000</a>
%e A375915 29539 is a term because 29539 = 109*271 is composite, 29539 == 9 (mod 10), and 5^((29539-1)/2) == 1 (mod 29539).
%o A375915 (PARI) isA375915(k) = (k>1) && !isprime(k) && (k%10==1 || k%10==9) && Mod(5,k)^((k-1)/2) == 1
%Y A375915                                    |        b=2        |   b=3   |   b=5    |
%Y A375915 -----------------------------------+-------------------+---------+----------+
%Y A375915   (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | this seq |
%Y A375915 -----------------------------------+-------------------+---------+----------+
%Y A375915  (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | A375916  |
%Y A375915 -----------------------------------+-------------------+---------+----------+
%Y A375915  b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490 | A375816  |
%Y A375915  (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |          |
%Y A375915 -----------------------------------+-------------------+---------+----------+
%Y A375915      Euler-Jacobi pseudoprimes     |      A047713      | A048950 | A375914  |
%Y A375915        (union of first two)        |                   |         |          |
%Y A375915 -----------------------------------+-------------------+---------+----------+
%Y A375915         Euler pseudoprimes         |      A006970      | A262051 | A262052  |
%Y A375915        (union of all three)        |                   |         |          |
%K A375915 nonn
%O A375915 1,1
%A A375915 _Jianing Song_, Sep 02 2024