A048950 Base-3 Euler-Jacobi pseudoprimes.

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

Author

Eric W. Weisstein

Keywords

• nonn

Comments

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]

Links

• 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

Crossrefs

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) | | | |

Programs

Mathematica

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

PARI

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