A191061 -id:A191061 - cp's OEIS frontend

A191017 Primes with Kronecker symbol (p|14) = 1.

3, 5, 13, 19, 23, 59, 61, 71, 79, 83, 101, 113, 127, 131, 137, 139, 151, 157, 173, 181, 191, 193, 227, 229, 233, 239, 251, 263, 269, 281, 283, 293, 307, 337, 349, 359, 397, 401, 419, 431, 449, 457, 461, 463, 467, 487, 509, 523, 563, 569, 587, 599, 617, 619

Offset: 1

T. D. Noe, May 24 2011

Originally incorrectly named "Primes that are squares mod 14", which is sequence A045373. - M. F. Hasler, Jan 15 2016
Conjecture: primes congruent to {1, 3, 5, 9, 13, 15, 19, 23, 25, 27, 39, 45} mod 56. - Vincenzo Librandi, Jun 22 2016
From Jianing Song, Nov 21 2019: (Start)
Rational primes that decompose in the field Q(sqrt(-14)).
These are primes p such that either one of (a) p == 1, 2, 4 (mod 7), p == 1, 7 (mod 8) or (b) p == 3, 5, 6 (mod 7), p == 3, 5 (mod 8) holds. So the conjecture above is correct. (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A191018, A191020, A191061, A274504.

[p: p in PrimesUpTo(619) | KroneckerSymbol(p, 14) eq 1]; // Vincenzo Librandi, Sep 11 2012

Select[Prime[Range[200]], JacobiSymbol[#,14]==1&]

is(p)=kronecker(p, 14)==1&&isprime(p) \\ Michel Marcus, Jun 23 2016 after A191032

Definition corrected (following an observation by David Broadhurst) by M. F. Hasler, Jan 15 2016

A377177 Primes p such that -7/2 is a primitive root modulo p.

Original entry on oeis.org

11, 17, 29, 31, 37, 41, 43, 47, 73, 89, 103, 107, 109, 149, 167, 179, 197, 257, 277, 311, 313, 317, 347, 353, 367, 373, 383, 389, 409, 433, 479, 491, 499, 503, 521, 541, 557, 571, 577, 593, 601, 607, 647, 653, 659, 683, 701, 719, 727, 761, 769, 821, 839, 857, 883, 887, 907, 929, 937, 947, 983

Offset: 1

Jianing Song, Oct 18 2024

If p is a term in this sequence, then -7/2 is not a square modulo p (i.e., p is in A191061).

Conjecture: this sequence has relative density equal to Artin's constant (A005596) with respect to the set of primes.

Robert Israel, Table of n, a(n) for n = 1..10000
Wikipedia, Artin's conjecture on primitive roots.
Index entries for primes by primitive root

Primes p such that +a/2 is a primitive root modulo p: A320384 (a=3), A377174 (a=5), A377176 (a=7), A377178 (a=9).
Primes p such that -a/2 is a primitive root modulo p: A377172 (a=3), A377175 (a=5), this sequence (a=7), A377179 (a=9).
Cf. A191061, A005596.

select(p -> isprime(p) and numtheory:-order(-7/2 mod p, p) = p-1, [seq(i,i=3..1000,2)]); # Robert Israel, Nov 10 2024

Cases[Prime[Range[2,170]],?(MemberQ[PrimitiveRootList[#],ResourceFunction["FractionMod"][-7/2,#]]&)] (* _Shenghui Yang, Oct 23 2024 *)

forprime(p=8,10^3,if(znorder(Mod(-7/2,p))==p-1,print1(p,", "))); \\ Joerg Arndt, Oct 19 2024

A274504 Primes with Kronecker symbol (p|14) != 1.

Original entry on oeis.org

2, 7, 11, 17, 29, 31, 37, 41, 43, 47, 53, 67, 73, 89, 97, 103, 107, 109, 149, 163, 167, 179, 197, 199, 211, 223, 241, 257, 271, 277, 311, 313, 317, 331, 347, 353, 367, 373, 379, 383, 389, 409, 421, 433, 439, 443, 479, 491, 499, 503, 521, 541, 547, 557

Offset: 1

Vincenzo Librandi, Jun 29 2016

Complement of A191017 in primes.

Conjecture: primes not congruent to (1, 3, 5, 9, 13, 15, 19, 23, 25, 27, 39, 45) mod 56.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A191017, A191061.

[p: p in PrimesUpTo(619) | not KroneckerSymbol(p, 14) eq 1];

Select[Prime[Range[200]], ! JacobiSymbol[#, 14] == 1 &]

A369863 Inert rational primes in the field Q(sqrt(-21)).

Original entry on oeis.org

13, 29, 43, 47, 53, 59, 61, 67, 73, 79, 83, 97, 113, 127, 131, 137, 149, 151, 157, 163, 167, 181, 197, 211, 227, 229, 233, 241, 251, 281, 311, 313, 317, 331, 349, 379, 383, 389, 397, 401, 409, 419, 433, 449, 463, 467, 479, 487, 499, 503, 547, 557, 563, 569, 571, 577, 587

Offset: 1

Dimitris Cardaris, Feb 03 2024

nonn

Primes p such that Legendre(-21,p) = -1.

Cf. inert rational primes in the imaginary quadratic field Q(sqrt(-d)) for the first squarefree positive integers d: A002145 (1), A003628 (2), A003627 (3), A003626 (5), A191059 (6), A003625 (7), A296925 (10), A191060 (11), A105885 (13), A191061 (14), A191062 (15), A296930 (17), A191063 (19), this sequence (21), A191064 (22), A191065 (23).

Select[Range[3,600], PrimeQ[#] && JacobiSymbol[-21,#]==-1 &] (* Stefano Spezia, Feb 04 2024 *)

[p for p in prime_range(3, 600) if legendre_symbol(-21, p) == -1]

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A191017 Primes with Kronecker symbol (p|14) = 1.

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A377177 Primes p such that -7/2 is a primitive root modulo p.

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A274504 Primes with Kronecker symbol (p|14) != 1.

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A369863 Inert rational primes in the field Q(sqrt(-21)).

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