A004538 -id:A004538 - cp's OEIS frontend

A038893 Odd primes p such that 21 is a square mod p.

3, 5, 7, 17, 37, 41, 43, 47, 59, 67, 79, 83, 89, 101, 109, 127, 131, 151, 163, 167, 173, 193, 211, 227, 251, 257, 269, 277, 293, 311, 331, 337, 353, 373, 379, 383, 419, 421, 457, 461, 463, 467, 479, 487, 499, 503

Offset: 1

N. J. A. Sloane

nonn

These primes correspond to the representation of the two classes of discriminant 21 of binary quadratic forms with principal reduced forms [1, 3, -3] and [3, 3, -1]. The first class represents the primes given in A141159 (or A139492). The second class gives the prime 3 (which divides 21), and primes congruent to 2 (mod 3) and also to 3, 5, 6 (mod 7). The solution of x^2 - 21 == 0 (mod p) leads to the representative primitive parallel forms for discriminant 21 and representation of primes p. - Wolfdieter Lang, Jun 19 2019

Prime factors of A082111 and excluding the 3, prime factors of A004538. - Klaus Purath, Jan 04 2023

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

Cf. A139492, A141159.

Select[Prime[Range[100]], JacobiSymbol[21, #] != -1 &] (* Vincenzo Librandi, Sep 07 2012 *)

isok(p) = (p>2) && isprime(p) && issquare(Mod(21, p)); \\ Michel Marcus, Jun 19 2019

Name clarified by Michel Marcus, Jun 22 2019

A318765 a(n) = (n + 2)*(n^2 + n - 1).

Original entry on oeis.org

-2, 3, 20, 55, 114, 203, 328, 495, 710, 979, 1308, 1703, 2170, 2715, 3344, 4063, 4878, 5795, 6820, 7959, 9218, 10603, 12120, 13775, 15574, 17523, 19628, 21895, 24330, 26939, 29728, 32703, 35870, 39235, 42804, 46583, 50578, 54795, 59240, 63919, 68838, 74003, 79420, 85095

Offset: 0

Bruno Berselli, Sep 04 2018

First differences are in A004538.

a(n) is divisible by 11 for n = 3, 7, 9, 14, 18, 20, 25, 29, 31, 36, 40, ... with formula (1/3)*(11*m + (1 + (m mod 3))*(-1)^((m-1) mod 3) + 8), m >= 0.

Colin Barker, Table of n, a(n) for n = 0..1000
Bruno Berselli, Table of similar sequences (row k=3, m>1).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A004538.
Subsequence of A047216.
Similar sequences (see Table in Links section): A011379, A027444, A033445, A034262, A045991, A069778.

GAP
```
List([0..50], n -> (n+2)*(n^2+n-1));
```

[(n+2)*(n^2+n-1) for n in 0:50] |> println

Magma
```
[(n+2)*(n^2+n-1): n in [0..50]];
```

seq((n+2)*(n^2+n-1),n=0..43); # Paolo P. Lava, Sep 04 2018

Table[(n + 2) (n^2 + n - 1), {n, 0, 50}]

Maxima
```
makelist((n+2)*(n^2+n-1), n, 0, 50);
```
PARI
```
vector(50, n, n--; (n+2)*(n^2+n-1))
```
Python
```
[(n+2)*(n**2+n-1) for n in range(50)]
```
Sage
```
[(n+2)*(n^2+n-1) for n in (0..50)]
```

O.g.f.: (-2 + 11*x - 4*x^2 + x^3)/(1 - x)^4.
E.g.f.: (-2 + 5*x + 6*x^2 + x^3)*exp(x).
a(n) = -A033445(-n-1).
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n >= 5. - Wesley Ivan Hurt, Dec 18 2020

cp's OEIS Frontend

A038893 Odd primes p such that 21 is a square mod p.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A318765 a(n) = (n + 2)*(n^2 + n - 1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Julia

Magma

Maple

Mathematica

Maxima

PARI

Python

Sage

Formula