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.

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

1,1

Comments

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

Links

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)

Example

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

Prog

(PARI) isA306310(k)=(k%8==5) && Mod(2, k)^((k-1)/2)==1
(PARI) isok(k) = (k>1) && (k%2) && (Mod(2, k)^((k-1)/2) == Mod(-kronecker(2, k), k)); \\ Michel Marcus, Feb 07 2019

Crossrefs

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) | | | |

Sequence in context: A143688 A086250 A285549 * A210454 A069309 A353021

Adjacent sequences: A306307 A306308 A306309 * A306311 A306312 A306313

Keyword

nonn

Author

Jianing Song, Feb 06 2019

Status

approved