A002383 -id:A002383 - cp's OEIS frontend

A147297 Primes of the form (2k)^2 + 3(2k + 1)^2.

31, 307, 463, 1123, 1723, 3307, 4831, 6007, 8011, 10303, 11131, 13807, 20023, 23563, 26083, 30103, 35911, 43891, 60271, 86143, 95791, 108571, 127807, 136531, 145543, 164431, 205663, 239611, 276151, 284623, 288907, 366631, 371491, 386263, 459007

Offset: 1

Kieren MacMillan, Nov 05 2008

nonn

First thirteen terms are a subset of A073337, A002383 and A085104.

[ a: n in [1..900] | IsPrime(a) where a is (2*n)^2 + 3*(2*n+1)^2] // Vincenzo Librandi, Nov 25 2010

makelist((2*k)^2+3*(2*k+1)^2,k,1,100)$ sublist(%,primep);

More terms from Vincenzo Librandi, Apr 28 2010

A237366 Primes p such that f(f(p)) is prime where f(x) = x^2+x+1.

Original entry on oeis.org

7, 19, 31, 67, 127, 181, 223, 241, 331, 367, 409, 463, 487, 673, 709, 751, 811, 823, 883, 997, 1117, 1231, 1321, 1489, 1549, 1861, 1933, 2083, 2179, 2287, 2473, 2551, 2707, 2803, 2851, 2857, 2917, 2971, 3067, 3361, 3499, 3559, 3691, 3847, 3931

Offset: 1

Derek Orr, Feb 06 2014

nonn

			31 is prime and (31^2+31+1)^2+(31^2+31+1)+1 = 987043 is prime. Thus, 31 is a member of this sequence.

Cf. A237360, A002383.

s=[]; forprime(p=2, 4000, if(isprime(p^4+2*p^3+4*p^2+3*p+3), s=concat(s, p))); s \\ Colin Barker, Feb 07 2014

import sympy
from sympy import isprime
{print(n) for n in range(10**4) if isprime(n) and isprime((n**2+n+1)**2+(n**2+n+1)+1)}

A306759 Decimal expansion of the sum of reciprocals of Brazilian primes, also called the Brazilian primes constant.

Original entry on oeis.org

3, 3, 1, 7, 5, 4, 4, 6, 6

Offset: 0

Bernard Schott, Mar 08 2019

The name "constant of Brazilian primes" is used in the article "Les nombres brésiliens" in link, théorème 4, page 36. Brazilian primes are in A085104.
Let S(k) be the sum of reciprocals of Brazilian primes < k. These values below come from different calculations by Jon, Michel, Daniel and Davis.
   q            S(10^q)
  ==    ========================
  0.1428571428571428571...  (= 1/7)
  0.2889927283868234859...
  0.3229022355626914481...
  0.3295236806353669357...
  0.3312171311946179843...
  0.3316038696349217289...
  0.3317139158654747333...
  0.3317434191078170412...
  0.3317513267394988538...
  0.3317535651668937256...
  0.3317542057931842329...
  0.3317543906772274268...
  0.3317544444033188051...
  0.3317544601136967527...
  0.3317544647354485208...
  0.3317544661014868080...
  0.3317544665073451951...
  0.3317544666282877863...
  0.3317544666644601817...
  0.3317544666753095766...
According to the Goormaghtigh conjecture, there are only two Brazilian primes which are twice Brazilian: 31 = (111)_5 = (11111)_2 and 8191 = (111)_90 = (1111111111111)_2. The reciprocals of these two numbers are counted only once in the sum.

			1/7 + 1/13 + 1/31 + 1/43 + 1/73 + 1/127 + 1/157 + ... = 0.33175...

Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, page 175.

Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
Wikipedia, Goormaghtigh conjecture.

Cf. A085104 (Brazilian primes), A002383 (Brazilian primes (111)_b), A225148 (Brazilian primes of the form (b^q-1)/(b-1) with q prime >= 5).

Cf. A173898 (sum of the reciprocals of the Mersenne primes), A065421 (Brun's constant).

brazil(N, L=List())=forprime(K=3, #binary(N+1)-1, for(n=2, sqrtnint(N-1, K-1), if(isprime((n^K-1)/(n-1)),listput(L, (n^K-1)/(n-1))))); Set(L);
brazilcons(lim,nbd) = r=brazil(10^lim); x=sum(M=1, #r, 1./r[M]);for(n=1, nbd, print1(floor(x*10^n)%10, ", "));\\ Davis Smith, Mar 10 2019

cons(lim)=my(v=List(), t, k); for(n=2, sqrt(lim), t=1+n; k=1; while((t+=n^k++)<=lim, if(isprime(t), listput(v, t)))); v = vecsort(Vec(v), , 8); sum(k=1, #v, 1./v[k]); \\ Michel Marcus, Mar 11 2019

Equals Sum_{n>=1} 1/A085104(n).

A308316 Numbers m such that q = 2^m - 1 and r = m^2 + m + 1 are both primes.

Original entry on oeis.org

2, 3, 5, 17, 89, 9689, 11213, 2976221

Offset: 1

Jaroslav Krizek, May 19 2019

All terms are primes.
Mersenne exponents p from A000043 such that p^2 + p + 1 is a prime.
Intersection of A000043 and A002384.
Corresponding values of primes q: 3, 7, 31, 131071, 618970019642690137449562111, ...
Corresponding values of primes r: 7, 13, 31, 307, 8011, 93886411, 125742583, 8857894417063, ...

Cf. A000043, A002383, A002384.

[m: m in [1..1000] | IsPrime(2^m - 1)  and IsPrime(m^2 + m + 1)]

isok(n) = isprime(2^n-1) && isprime(n^2+n+1); \\ Michel Marcus, May 21 2019

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A147297 Primes of the form (2k)^2 + 3(2k + 1)^2.

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A237366 Primes p such that f(f(p)) is prime where f(x) = x^2+x+1.

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A306759 Decimal expansion of the sum of reciprocals of Brazilian primes, also called the Brazilian primes constant.

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A308316 Numbers m such that q = 2^m - 1 and r = m^2 + m + 1 are both primes.

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Magma

PARI