A296937 Rational primes that decompose in the field Q(sqrt(13)).

3, 17, 23, 29, 43, 53, 61, 79, 101, 103, 107, 113, 127, 131, 139, 157, 173, 179, 181, 191, 199, 211, 233, 251, 257, 263, 269, 277, 283, 311, 313, 337, 347, 367, 373, 389, 419, 433, 439, 443, 467, 491, 503, 521, 523, 547, 563, 569, 571, 599, 601, 607, 641

Offset: 1

Author

N. J. A. Sloane, Dec 26 2017

Keywords

• nonn
• easy

Comments

Is this the same sequence as A141188 or A038883? - R. J. Mathar, Jan 02 2018

From Jianing Song, Apr 21 2022: (Start)

Primes p such that Kronecker(13, p) = Kronecker(p, 13) = 1, where Kronecker() is the Kronecker symbol. That is to say, primes p that are quadratic residues modulo 13.

Primes p such that p^6 == 1 (mod 13).

Primes p == 1, 3, 4, 9, 10, 12 (mod 13). (End)

Links

• Jianing Song, Table of n, a(n) for n = 1..10000
• Index to sequences related to decomposition of primes in quadratic fields

Crossrefs

Cf. A011583 (kronecker symbol modulo 13), A038883.

Rational primes that decompose in the quadratic field with discriminant D: A139513 (D=-20), A191019 (D=-19), A191018 (D=-15), A296920 (D=-11), A033200 (D=-8), A045386 (D=-7), A002144 (D=-4), A002476 (D=-3), A045468 (D=5), A001132 (D=8), A097933 (D=12), this sequence (D=13), A296938 (D=17).

Cf. A038884 (inert rational primes in the field Q(sqrt(13))).

Programs

Maple

Load the Maple program HH given in A296920. Then run HH(13, 200); This produces A296937, A038884, A038883.

PARI

isA296937(p) = isprime(p) && kronecker(p,13) == 1 \\ Jianing Song, Apr 21 2022

Formula

Equals A038883 \ {13}. - Jianing Song, Apr 21 2022