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

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

1,1

Comments

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, ...

Links

Table of n, a(n) for n=1..37.

Example

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

Prog

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

Crossrefs

| 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), | | | |

conjectured to be 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) | | | |

Sequence in context: A206466 A137434 A175111 * A115190 A231706 A372964

Adjacent sequences: A375914 A375915 A375916 * A375918 A375919 A375920

Keyword

nonn,new

Author

Jianing Song, Sep 02 2024

Status

approved