A244626 -id:A244626 - cp's OEIS frontend

A047713 Euler-Jacobi pseudoprimes: 2^((n-1)/2) == (2 / n) mod n, where (2 / n) is a Jacobi symbol.

561, 1105, 1729, 1905, 2047, 2465, 3277, 4033, 4681, 6601, 8321, 8481, 10585, 12801, 15841, 16705, 18705, 25761, 29341, 30121, 33153, 34945, 41041, 42799, 46657, 49141, 52633, 62745, 65281, 74665, 75361, 80581, 85489, 87249, 88357, 90751, 104653

Offset: 1

N. J. A. Sloane, Richard Pinch and Robert G. Wilson v

Odd composite numbers n such that 2^((n-1)/2) == (-1)^((n^2-1)/8) mod n. - Thomas Ordowski, Dec 21 2013

Most terms are congruent to 1 mod 8 (cf. A006971). Among the first 1000 terms, the number of terms congruent to 1, 3, 5 and 7 mod 8 are 764, 47, 125 and 64, respectively. - Jianing Song, Sep 05 2018

R. K. Guy, Unsolved Problems in Number Theory, A12.
H. Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes the subsequence A006971).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Euler-Jacobi Pseudoprime.
Eric Weisstein's World of Mathematics, Pseudoprime.
Index entries for sequences related to pseudoprimes

Cf. A002997, A001567, A048950.

Terms in this sequence satisfying certain congruence: A270698 (congruent to 1 mod 4), A270697 (congruent to 3 mod 4), A006971 (congruent to +-1 mod 8), A244628 (congruent to 3 mod 8), A244626 (congruent to 5 mod 8).

Select[ Range[ 3, 105000, 2 ], Mod[ 2^((# - 1)/2) - JacobiSymbol[ 2, # ], # ] == 0 && ! PrimeQ[ # ] & ]

is(n)=n%2 && Mod(2,n)^(n\2)==kronecker(2,n) && !isprime(n) \\ Charles R Greathouse IV, Dec 20 2013

Corrected by Eric W. Weisstein; more terms from David W. Wilson

A006971 Composite numbers k such that k == +-1 (mod 8) and 2^((k-1)/2) == 1 (mod k).

Original entry on oeis.org

561, 1105, 1729, 1905, 2047, 2465, 4033, 4681, 6601, 8321, 8481, 10585, 12801, 15841, 16705, 18705, 25761, 30121, 33153, 34945, 41041, 42799, 46657, 52633, 62745, 65281, 74665, 75361, 85489, 87249, 90751, 113201, 115921, 126217, 129921, 130561, 149281, 158369

Offset: 1

Richard Pinch

nonn

Previous name was "Terms of A047713 that are congruent to +-1 mod 8".

Complement of (A244626 union A244628) with respect to A047713. - Jianing Song, Sep 18 2018

Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section A12, pp. 44-50.
Hans Riesel, Prime numbers and computer methods for factorization, Progress in Mathematics, Vol. 57, Birkhäuser, Boston, 1985.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..828 from Jianing Song using data from A047713)

Subsequence of A001567 and A047713.

Cf. A244626, A244628, A270697, A270698.

Select[Range[10^5], MemberQ[{1, 7}, Mod[#, 8]] && CompositeQ[#] && PowerMod[2, (# - 1)/2, #] == 1 &] (* Amiram Eldar, Nov 06 2023 *)

This sequence appeared as M5461 in Sloane-Plouffe (1995), but was later mistakenly declared to be an erroneous form of A047713. Thanks to Jianing Song for providing the correct definition. - N. J. A. Sloane, Sep 17 2018

Formal definition by Jianing Song, Sep 18 2018

A048950 Base-3 Euler-Jacobi pseudoprimes.

Original entry on oeis.org

121, 703, 1729, 1891, 2821, 3281, 7381, 8401, 8911, 10585, 12403, 15457, 15841, 16531, 18721, 19345, 23521, 24661, 28009, 29341, 31621, 41041, 44287, 46657, 47197, 49141, 50881, 52633, 55969, 63139, 63973, 74593, 75361, 79003, 82513

Offset: 1

Eric W. Weisstein

nonn

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

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]

Amiram Eldar, Table of n, a(n) for n = 1..10000
A. Rotkiewicz, On Euler Lehmer pseudoprimes and strong Lehmer pseudoprimes with Parameters L, Q in arithmetic progressions, Math. Comp 39 (159) (1982) 239-247.
Eric Weisstein's World of Mathematics, Euler-Jacobi Pseudoprime.
Index entries for sequences related to pseudoprimes

Cf. A005935.
                                   |        b=2        |   b=3    |   b=5   |
-----------------------------------+-------------------+----------+---------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917  | A375915 |
-----------------------------------+-------------------+----------+---------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918  | A375916 |
-----------------------------------+-------------------+----------+---------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490  | A375816 |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |          |         |
-----------------------------------+-------------------+----------+---------+
     Euler-Jacobi pseudoprimes     |      A047713      | this seq | A375914 |
       (union of first two)        |                   |          |         |
-----------------------------------+-------------------+----------+---------+
        Euler pseudoprimes         |      A006970      | A262051  | A262052 |
       (union of all three)        |                   |          |         |

Select[Range[1, 10^5, 2], GCD[#, 3] == 1 && CompositeQ[#] && PowerMod[3, (# - 1)/2, #] == Mod[JacobiSymbol[3, #], #] &] (* Amiram Eldar, Jun 28 2019 *)

is(n) = n%2==1 && gcd(n,3)==1 && Mod(3, n)^((n-1)/2)==kronecker(3,n)
forcomposite(c=1, 83000, if(is(c), print1(c, ", "))) \\ Felix Fröhlich, Jul 15 2019

A306310 Odd numbers k > 1 such that 2^((k-1)/2) == -(2/k) = -A091337(k) (mod k), where (2/k) is the Jacobi (or Kronecker) symbol.

Original entry on oeis.org

341, 5461, 10261, 15709, 31621, 49981, 65077, 83333, 137149, 176149, 194221, 215749, 219781, 276013, 282133, 534061, 587861, 611701, 653333, 657901, 665333, 688213, 710533, 722261, 738541, 742813, 769757, 950797, 1064053, 1073021, 1109461, 1141141, 1357621, 1398101

Offset: 1

Jianing Song, Feb 06 2019

nonn

All terms are composite because for odd primes p we always have 2^((p-1)/2) == (2/p) = A091337(p) (mod p).
Note that if k is odd and b^((k-1)/2) == -(b/k) (mod k), then taking Jacobi symbol modulo k (which depends only on the remainder modulo k) yields (b/k)^((k-1)/2) = -(b/k), or (b/k)^((k+1)/2) = -1. This implies that (k+1)/2 is odd, so k == 1 (mod 4). Moreover, if k > 1, then (b/k) = -1 (see the Math Stack Exchange link below), so b^((k-1)/2) == 1 (mod k). In particular, this sequence is equivalent to "numbers k == 5 (mod 8) such that 2^((k-1)/2) == 1 (mod k)". [Comment rewritten by Jianing Song, Sep 07 2024]
Also numbers k in A001567 and congruent to 5 modulo 8 such that k - 1 divided by the multiplicative order of 2 modulo k is an even number.
Euler pseudoprimes (A006970) that are not Euler-Jacobi pseudoprimes (A047713). - Amiram Eldar, Oct 28 2019

			341 is a term because (2/341) = -1, and 2^((341-1)/2) == 1 (mod 341).

Amiram Eldar, Table of n, a(n) for n = 1..1000
Mathematics Stack Exchange, There are no a in Z and odd k > 1 such that (a/k) = 1 and a^((k-1)/2) == -1 (mod k)

Cf. A091337, A001567, A007814.
                                   |        b=2        |   b=3   |   b=5   |
-----------------------------------+-------------------+---------+---------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | A375915 |
-----------------------------------+-------------------+---------+---------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | A375916 |
-----------------------------------+-------------------+---------+---------+
 b^((k-1)/2)==-(b/k) (mod k), also |      this seq     | A375490 | A375816 |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |         |
-----------------------------------+-------------------+---------+---------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950 | A375914 |
       (union of first two)        |                   |         |         |
-----------------------------------+-------------------+---------+---------+
        Euler pseudoprimes         |      A006970      | A262051 | A262052 |
       (union of all three)        |                   |         |         |

isA306310(k)=(k%8==5) && Mod(2, k)^((k-1)/2)==1

isok(k) = (k>1) && (k%2) && (Mod(2, k)^((k-1)/2) == Mod(-kronecker(2, k), k)); \\ Michel Marcus, Feb 07 2019

A375490 Odd numbers k > 1 such that gcd(3,k) = 1 and 3^((k-1)/2) == -(3/k) (mod k), where (3/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 3 (A262051) that are not Euler-Jacobi pseudoprimes to base 3 (A048950).

Original entry on oeis.org

1541, 2465, 4961, 30857, 31697, 72041, 83333, 162401, 192713, 206981, 258017, 359369, 544541, 565001, 574397, 653333, 929633, 1018601, 1032533, 1133441, 1351601, 1373633, 1904033, 1953281, 2035661, 2797349, 2864501, 3264797, 3375041, 3554633, 3562361, 3636161

Offset: 1

Jianing Song, Sep 01 2024

nonn

Note that if k is odd and b^((k-1)/2) == -(b/k) (mod k), then taking Jacobi symbol modulo k (which depends only on the remainder modulo k) yields (b/k)^((k-1)/2) = -(b/k), or (b/k)^((k+1)/2) = -1. This implies that (k+1)/2 is odd, so k == 1 (mod 4). Moreover, if k > 1, then (b/k) = -1 (see the Math Stack Exchange link below), so b^((k-1)/2) == 1 (mod k). In particular, this sequence is equivalent to "numbers k == 5 (mod 12) such that 3^((k-1)/2) == 1 (mod k)". [Comment rewritten by Jianing Song, Sep 07 2024]

			1541 is a term because (3/1541) = -1, and 3^((1541-1)/2) == 1 (mod 1541).

Jianing Song, Table of n, a(n) for n = 1..1000
Mathematics Stack Exchange, There are no a in Z and odd k > 1 such that (a/k) = 1 and a^((k-1)/2) == -1 (mod k)

                                   |        b=2        |   b=3    |   b=5   |
-----------------------------------+-------------------+----------+---------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917  | A375915 |
-----------------------------------+-------------------+----------+---------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918  | A375916 |
-----------------------------------+-------------------+----------+---------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | this seq | A375816 |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |          |         |
-----------------------------------+-------------------+----------+---------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950  | A375914 |
       (union of first two)        |                   |          |         |
-----------------------------------+-------------------+----------+---------+
        Euler pseudoprimes         |      A006970      | A262051  | A262052 |
       (union of all three)        |                   |          |         |

isA375490(k) = (k>1) && gcd(k,6)==1 && Mod(3,k)^((k-1)/2)==-kronecker(3,k)

isA375490(k) = k%12==5 && Mod(3,k)^((k-1)/2)==1 \\ Jianing Song, Sep 07 2024

A375816 Odd numbers k > 1 such that gcd(5,k) = 1 and 5^((k-1)/2) == -(5/k) (mod k), where (5/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 5 (A262052) that are not Euler-Jacobi pseudoprimes to base 5 (A375914).

Original entry on oeis.org

217, 13333, 16297, 23653, 30673, 44173, 46657, 48133, 56033, 98173, 130417, 131977, 136137, 179893, 188113, 190513, 197633, 267977, 334153, 334657, 347777, 360533, 407353, 412933, 421637, 486157, 667153, 670033, 677917, 694153, 710533, 765073, 839833, 935137, 997633

Offset: 1

Jianing Song, Sep 01 2024

nonn

Note that if k is odd and b^((k-1)/2) == -(b/k) (mod k), then taking Jacobi symbol modulo k (which depends only on the remainder modulo k) yields (b/k)^((k-1)/2) = -(b/k), or (b/k)^((k+1)/2) = -1. This implies that (k+1)/2 is odd, so k == 1 (mod 4). Moreover, if k > 1, then (b/k) = -1 (see the Math Stack Exchange link below), so b^((k-1)/2) == 1 (mod k). In particular, this sequence is equivalent to "numbers k == 13, 17 (mod 20) such that 5^((k-1)/2) == 1 (mod k)". [Comment rewritten by Jianing Song, Sep 07 2024]

			217 is a term because (5/217) = -1, and 5^((217-1)/2) == 1 (mod 217).

Jianing Song, Table of n, a(n) for n = 1..1000
Mathematics Stack Exchange, There are no a in Z and odd k > 1 such that (a/k) = 1 and a^((k-1)/2) == -1 (mod k)

                                   |        b=2        |   b=3   |   b=5    |
-----------------------------------+-------------------+---------+----------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | A375915  |
-----------------------------------+-------------------+---------+----------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | A375916  |
-----------------------------------+-------------------+---------+----------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490 | this seq |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |          |
-----------------------------------+-------------------+---------+----------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950 | A375914  |
       (union of first two)        |                   |         |          |
-----------------------------------+-------------------+---------+----------+
        Euler pseudoprimes         |      A006970      | A262051 | A262052  |
       (union of all three)        |                   |         |          |

isA375816(k) = (k>1) && gcd(k,10)==1 && Mod(5,k)^((k-1)/2)==-kronecker(5,k)

isA375816(k) = (k%20==13 || k%20==17) && Mod(5,k)^((k-1)/2)==1

A375914 Base-5 Euler-Jacobi pseudoprimes: odd composite k coprime to 5 such that 5^((k-1)/2) == (5/k) (mod n), where (5/k) is the Jacobi symbol (or Kronecker symbol).

Original entry on oeis.org

781, 1541, 1729, 5461, 5611, 6601, 7449, 7813, 11041, 12801, 13021, 14981, 15751, 15841, 21361, 24211, 25351, 29539, 38081, 40501, 41041, 44801, 47641, 53971, 67921, 75361, 79381, 90241, 100651, 102311, 104721, 106201, 106561, 112141, 113201, 115921, 121463, 133141

Offset: 1

Jianing Song, Sep 02 2024

nonn

			781 is a term because 781 = 11*71 is composite, (5/781) = 1, and 5^((781-1)/2) == 1 (mod 781).
7813 is a term because 7813 = 13*601 is composite, (5/7813) = -1, and 5^((7813-1)/2) == -1 (mod 7813).

Jianing Song, Table of n, a(n) for n = 1..1000

                                   |        b=2        |   b=3   |   b=5    |
-----------------------------------+-------------------+---------+----------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | A375915  |
-----------------------------------+-------------------+---------+----------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | A375916  |
-----------------------------------+-------------------+---------+----------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490 | A375816  |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |          |
-----------------------------------+-------------------+---------+----------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950 | this seq |
       (union of first two)        |                   |         |          |
-----------------------------------+-------------------+---------+----------+
        Euler pseudoprimes         |      A006970      | A262051 | A262052  |
       (union of all three)        |                   |         |          |

isA375914(k) = k>1 && !isprime(k) && gcd(k,10)==1 && Mod(5,k)^((k-1)/2)==kronecker(5,k)

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

Original entry on oeis.org

781, 1541, 1729, 5461, 5611, 6601, 7449, 11041, 12801, 13021, 14981, 15751, 15841, 21361, 24211, 25351, 29539, 38081, 40501, 41041, 44801, 47641, 53971, 67921, 75361, 79381, 90241, 100651, 102311, 104721, 106201, 106561, 112141, 113201, 115921, 133141, 135201, 141361

Offset: 1

Jianing Song, Sep 02 2024

nonn

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).

			29539 is a term because 29539 = 109*271 is composite, 29539 == 9 (mod 10), and 5^((29539-1)/2) == 1 (mod 29539).

Jianing Song, Table of n, a(n) for n = 1..1000

                                   |        b=2        |   b=3   |   b=5    |
-----------------------------------+-------------------+---------+----------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | this seq |
-----------------------------------+-------------------+---------+----------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | A375916  |
-----------------------------------+-------------------+---------+----------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490 | A375816  |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |          |
-----------------------------------+-------------------+---------+----------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950 | A375914  |
       (union of first two)        |                   |         |          |
-----------------------------------+-------------------+---------+----------+
        Euler pseudoprimes         |      A006970      | A262051 | A262052  |
       (union of all three)        |                   |         |          |

isA375915(k) = (k>1) && !isprime(k) && (k%10==1 || k%10==9) && Mod(5,k)^((k-1)/2) == 1

A375916 Composite numbers k == 3, 7 (mod 10) such that 5^((k-1)/2) == -1 (mod k).

Original entry on oeis.org

7813, 121463, 195313, 216457, 315283, 319507, 353827, 555397, 559903, 753667, 939727, 1164083, 1653667, 1663213, 1703677, 1809697, 1958503, 2255843, 2339377, 2423323, 2942333, 2987167, 3313643, 4265257, 4635053, 5376463, 5979247, 6611977, 7784297, 7859707

Offset: 1

Jianing Song, Sep 02 2024

nonn

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).

			216457 is a term because 216457 = 233*929 is a composite, 216457 == 7 (mod 10), and 5^((216457-1)/2) == -1 (mod 216457).

Jianing Song, Table of n, a(n) for n = 1..1000

                                   |        b=2        |   b=3   |   b=5    |
-----------------------------------+-------------------+---------+----------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | A375917 | A375915  |
-----------------------------------+-------------------+---------+----------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918 | this seq |
-----------------------------------+-------------------+---------+----------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490 | A375816  |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |         |          |
-----------------------------------+-------------------+---------+----------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950 | A375914  |
       (union of first two)        |                   |         |          |
-----------------------------------+-------------------+---------+----------+
        Euler pseudoprimes         |      A006970      | A262051 | A262052  |
       (union of all three)        |                   |         |          |

isA375916(k) = !isprime(k) && (k%10==3 || k%10==7) && Mod(5,k)^((k-1)/2) == -1

A375917 Composite numbers k == 1, 11 (mod 12) such that 3^((k-1)/2) == 1 (mod k).

Original entry on oeis.org

121, 1729, 2821, 7381, 8401, 10585, 15457, 15841, 18721, 19345, 23521, 24661, 28009, 29341, 31621, 41041, 46657, 47197, 49141, 50881, 52633, 55969, 63973, 74593, 75361, 82513, 87913, 88573, 93961, 111361, 112141, 115921, 125665, 126217, 138481, 148417, 172081

Offset: 1

Jianing Song, Sep 02 2024

nonn

Odd composite numbers k such that 3^((k-1)/2) == (3/k) = 1 (mod k), where (3/k) is the Jacobi symbol (or Kronecker symbol).

It seems that most terms are congruent to 1 modulo 12. The first terms congruent to 11 modulo 12 are 1683683, 1898999, 2586083, 2795519, 4042403, 4099439, 5087171, 8243111, ...

			1683683 is a term because 1683683 = 59*28537 is composite, 1683683 == 11 (mod 12), and 3^((1683683-1)/2) == 1 (mod 1683683).

Jianing Song, Table of n, a(n) for n = 1..1000

                                   |        b=2        |   b=3    |   b=5   |
-----------------------------------+-------------------+----------+---------+
  (b/k)=1, b^((k-1)/2)==1 (mod k)  |      A006971      | this seq | A375915 |
-----------------------------------+-------------------+----------+---------+
 (b/k)=-1, b^((k-1)/2)==-1 (mod k) | A244628 U A244626 | A375918  | A375916 |
-----------------------------------+-------------------+----------+---------+
 b^((k-1)/2)==-(b/k) (mod k), also |      A306310      | A375490  | A375816 |
 (b/k)=-1, b^((k-1)/2)==1 (mod k)  |                   |          |         |
-----------------------------------+-------------------+----------+---------+
     Euler-Jacobi pseudoprimes     |      A047713      | A048950  | A375914 |
       (union of first two)        |                   |          |         |
-----------------------------------+-------------------+----------+---------+
        Euler pseudoprimes         |      A006970      | A262051  | A262052 |
       (union of all three)        |                   |          |         |

isA375917(k) = (k>1) && !isprime(k) && (k%12==1 || k%12==11) && Mod(3,k)^((k-1)/2) == 1

cp's OEIS Frontend

A047713 Euler-Jacobi pseudoprimes: 2^((n-1)/2) == (2 / n) mod n, where (2 / n) is a Jacobi symbol.

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A006971 Composite numbers k such that k == +-1 (mod 8) and 2^((k-1)/2) == 1 (mod k).

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A048950 Base-3 Euler-Jacobi pseudoprimes.

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A306310 Odd numbers k > 1 such that 2^((k-1)/2) == -(2/k) = -A091337(k) (mod k), where (2/k) is the Jacobi (or Kronecker) symbol.

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A375490 Odd numbers k > 1 such that gcd(3,k) = 1 and 3^((k-1)/2) == -(3/k) (mod k), where (3/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 3 (A262051) that are not Euler-Jacobi pseudoprimes to base 3 (A048950).

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PARI

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A375816 Odd numbers k > 1 such that gcd(5,k) = 1 and 5^((k-1)/2) == -(5/k) (mod k), where (5/k) is the Jacobi symbol (or Kronecker symbol); Euler pseudoprimes to base 5 (A262052) that are not Euler-Jacobi pseudoprimes to base 5 (A375914).

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PARI

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A375914 Base-5 Euler-Jacobi pseudoprimes: odd composite k coprime to 5 such that 5^((k-1)/2) == (5/k) (mod n), where (5/k) is the Jacobi symbol (or Kronecker symbol).

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

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