A191059 -id:A191059 - cp's OEIS frontend

A377172 Primes p such that -3/2 is a primitive root modulo p.

17, 23, 37, 41, 43, 47, 67, 89, 109, 113, 137, 139, 157, 163, 167, 191, 229, 233, 239, 257, 263, 277, 283, 311, 349, 353, 359, 379, 383, 397, 421, 449, 479, 503, 521, 523, 541, 547, 569, 571, 593, 599, 613, 619, 641, 647, 661, 719, 733, 739, 743, 757, 761, 787, 809, 811, 839, 853, 857, 859, 863, 877, 887, 911, 929, 953, 977, 983

Offset: 1

Jianing Song, Oct 18 2024

If p is a term in this sequence, then -3/2 is not a square modulo p (i.e., p is in A191059).

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

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: this sequence (a=3), A377175 (a=5), A377177 (a=7), A377179 (a=9).
Cf. A191059, A005596.

forprime(p=5, 10^3, if(znorder(Mod(-3/2, p))==p-1, print1(p, ", ")));

A296924 Primes p such that Legendre(-6,p) = 0 or 1.

Original entry on oeis.org

2, 3, 5, 7, 11, 29, 31, 53, 59, 73, 79, 83, 97, 101, 103, 107, 127, 131, 149, 151, 173, 179, 193, 197, 199, 223, 227, 241, 251, 269, 271, 293, 313, 317, 337, 347, 367, 389, 409, 419, 433, 439, 443, 457, 461, 463, 467, 487, 491, 509, 557, 563, 577, 587, 601, 607, 631, 653, 659, 673, 677, 683

Offset: 1

N. J. A. Sloane, Dec 26 2017

nonn

Primes == 1, 2, 3, 5, 7, or 11 (mod 24). - Robert Israel, Dec 27 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A157437, A191059.

Load the Maple program HH given in A296920. Then run HH(-6, 200); This produces A157437, A191059, and the present sequence.
select(isprime, {seq(seq(24*i+j,j=[1,2,3,5,7,11]),i=0..100)});

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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A377172 Primes p such that -3/2 is a primitive root modulo p.

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A296924 Primes p such that Legendre(-6,p) = 0 or 1.

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

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