A191065 -id:A191065 - cp's OEIS frontend

A256567 Primes p with the property that there are three consecutive integers (x,x+1,x+2) with 1 < x <= p-3 whose product is 1 modulo p.

7, 11, 17, 19, 23, 37, 43, 53, 59, 61, 67, 79, 83, 89, 97, 101, 103, 107, 109, 113, 137, 149, 157, 167, 173, 181, 191, 199, 211, 223, 227, 229, 241, 251, 263, 271, 281, 283, 293, 307, 313, 317, 337, 347, 359, 367, 373, 379, 383, 389, 401, 419, 421, 431, 433, 449

Offset: 1

Marian Kraus, Apr 02 2015

nonn

There may be one or more such triples, but 23 is the only prime up to 100000 having precisely two such triples. For the number of triples for each prime, see A256572.

Together with 5, supersequence of A191065. - Arkadiusz Wesolowski, Nov 24 2021

			For p=7: 4*5*6=120==1 (mod 7), so 7 is a term.
For p=11: 5*6*7=210==1 (mod 11), so 11 is a term.
For p=17: 4*5*6=120==1 (mod 17), so 17 is a term.
13 is not a term because there is no such triple with product ==  1 (mod 13).

Marian Kraus, Table of n, a(n) for n = 1..6390

Cf. A191065, A256572, A254678, A256580.

isok(p) = {if (isprime(p), for (x=1, p-3, if (Mod(x*(x+1)*(x+2), p) == 1, return (1));););} \\ Michel Marcus, Oct 05 2021

library(numbers)
IP <- vector()
t <- vector()
S <- vector()
IP <- c(Primes(1000)) # Build a vector of all primes < 1000.
for (j in 1:(length(IP))){
   for (i in 3:(IP[j]-2))
      t[i-1] <- as.vector(mod(((i-1)*i*(i+1)),IP[j]))
   S[j] <- length(which(t==1))
}
IP[S!=0]
#The loop checks for every triple for every prime, what it is modulo that prime. "IP[S!=0]" lists the primes that have at least one triple. For all p<10000 it takes a few minutes. For all p<100000 a few hours.

A278579 Quadratic non-residues of 23: numbers n such that Jacobi(n,23) = -1.

Original entry on oeis.org

5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22, 28, 30, 33, 34, 37, 38, 40, 42, 43, 44, 45, 51, 53, 56, 57, 60, 61, 63, 65, 66, 67, 68, 74, 76, 79, 80, 83, 84, 86, 88, 89, 90, 91, 97, 99, 102, 103, 106, 107, 109, 111, 112, 113, 114, 120, 122, 125, 126, 129, 130, 132, 134, 135, 136, 137, 143, 145, 148, 149

Offset: 1

N. J. A. Sloane, Nov 29 2016

nonn

Important for the study of Ramanujan numbers A000594.

Wilton, John Raymond. "Congruence properties of Ramanujan's function τ(n)." Proceedings of the London Mathematical Society 2.1 (1930): 1-10. See page 1.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A028736, A000594, A063987, A278580, A028759 (=first 22 terms).

For the primes in this sequence see A191065.

LinearRecurrence[{1,0,0,0,0,0,0,0,0,0,1,-1},{5,7,10,11,14,15,17,19,20,21,22,28},80] (* Harvey P. Dale, Jan 12 2020 *)

From Robert Israel, Nov 30 2016: (Start)
a(n+11) = a(n)+23.
G.f.: (x^11+x^10+x^9+x^8+2*x^7+2*x^6+x^5+3*x^4+x^3+3*x^2+2*x+5)/(x^12-x^11-x+1). (End)

A035167 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = -23.

Original entry on oeis.org

1, 2, 2, 3, 0, 4, 0, 4, 3, 0, 0, 6, 2, 0, 0, 5, 0, 6, 0, 0, 0, 0, 1, 8, 1, 4, 4, 0, 2, 0, 2, 6, 0, 0, 0, 9, 0, 0, 4, 0, 2, 0, 0, 0, 0, 2, 2, 10, 1, 2, 0, 6, 0, 8, 0, 0, 0, 4, 2, 0, 0, 4, 0, 7, 0, 0, 0, 0, 2, 0, 2, 12, 2, 0, 2, 0, 0, 8, 0, 0, 5

Offset: 1

N. J. A. Sloane

Amiram Eldar, Table of n, a(n) for n = 1..10000
LMFDB, Dedekind zeta-function: zeta_K(s), where K is the number field with defining polynomial x^2-x+6.

Cf. A191021, A191065.

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -23, d], { d, Divisors[ n]}]]; (* Michael Somos, Jan 24 2021 *)

my(m = -23); direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -23, d)))}; /* Michael Somos, Jan 24 2021 */

From Amiram Eldar, Nov 17 2023: (Start)
a(n) = Sum_{d|n} Kronecker(-23, d).
Multiplicative with a(23^e) = 1, a(p^e) = (1+(-1)^e)/2 if Kronecker(-23, p) = -1 (p is in A191065), and a(p^e) = e+1 if Kronecker(-23, p) = 1 (p is in A191021).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3*Pi/sqrt(23) = 1.965202... . (End)

A296932 Primes p such that Legendre(-23,p) = 0 or 1.

Original entry on oeis.org

2, 3, 13, 23, 29, 31, 41, 47, 59, 71, 73, 101, 127, 131, 139, 151, 163, 167, 173, 179, 193, 197, 211, 223, 233, 239, 257, 269, 271, 277, 307, 311, 317, 331, 347, 349, 353, 397, 409, 439, 443, 449, 461, 463, 487, 491, 499, 509, 541, 547, 577, 587

Offset: 1

N. J. A. Sloane, Dec 26 2017

nonn

Primes == 0, 1, 2, 3, 4, 6, 8, 9, 12, 13, 16, or 18 (mod 23). - Robert Israel, Dec 26 2017

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

Load the Maple program HH given in A296920. Then run HH(-23, 200); This produces A191021, A191065, A296932.
Alternative:
select(isprime, [2,seq(seq(46*i+j,j=[1, 3, 9, 13, 23, 25, 27, 29, 31, 35, 39, 41]),i=0..30)]); # Robert Israel, Dec 26 2017

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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A256567 Primes p with the property that there are three consecutive integers (x,x+1,x+2) with 1 < x <= p-3 whose product is 1 modulo p.

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A278579 Quadratic non-residues of 23: numbers n such that Jacobi(n,23) = -1.

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A035167 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -23.

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A296932 Primes p such that Legendre(-23,p) = 0 or 1.

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

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A035167 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = -23.