A225148 -id:A225148 - cp's OEIS frontend

A002383 Primes of form k^2 + k + 1.

3, 7, 13, 31, 43, 73, 157, 211, 241, 307, 421, 463, 601, 757, 1123, 1483, 1723, 2551, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9901, 10303, 11131, 12211, 12433, 13807, 14281, 17293, 19183, 20023, 20593, 21757, 22651, 23563

Offset: 1

N. J. A. Sloane

Also primes p such that 4p-3 is square. - Giovanni Teofilatto, Sep 07 2005
Also these primes are sums of 1 and some consecutive even numbers starting at 2; e.g., 31 = 1+2+4+6+8+10. - Labos Elemer, Apr 15 2003
Also primes of form n^2 - n + 1 (Prime central polygonal numbers, A002061). - Zak Seidov, Jan 26 2006
Also primes which are of the form TriangularNumber(n) + TriangularNumber(n+2): 7 = 1+6, 13 = 3+10, 31 = 10+21, 43 = 15+28, 73 = 28+45, ... - Vladimir Joseph Stephan Orlovsky, Apr 03 2009
It is not known whether there are infinitely many primes of the form n^2+n+1. See Rose reference. - Daniel Tisdale, Jun 27 2009
These numbers when >= 7 are prime repunits 111_n in a base n >= 2, so except for 3, they are all Brazilian primes belonging to A085104. (See Links "Les nombres brésiliens", Sections V.4 - V.5.) A002383 is generated by A002384 which lists the bases n of 111_n. A002383 = A053183 Union A185632. - Bernard Schott, Dec 22 2012
Conjecture: the set of these numbers, except 3, is the intersection of sets A085104 and A059055. See A225148. - Thomas Ordowski, May 02 2013
For a(n)>13, the fractional part of square root of a(n) starts with digit 5 (see A034101). - Charles Kusniec, Sep 06 2022

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 46.
L. Poletti, Le serie dei numeri primi appartenente alle due forme quadratiche (A) n^2+n+1 e (B) n^2+n-1 per l'intervallo compreso entro 121 milioni, e cioè per tutti i valori di n fino a 11000, Atti della Reale Accademia Nazionale dei Lincei, Memorie della Classe di Scienze Fisiche, Matematiche e Naturali, s. 6, v. 3 (1929), pages 193-218.
H. E. Rose, A Course in Number Theory, Clarendon Press, 1988, p. 217.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Zak Seidov, Table of n, a(n) for n = 1..10751
Cody S. Hansen and Pace P. Nielsen, Prime factors of phi3(x) of the same form, arXiv:2204.08971 [math.NT], 2022.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A002384, A088503, A110284, A085104.
Cf. A237037, A237038, A237039, A237040 (from semiprimes of form n^3 + 1).
See also A034101.

[ a: n in [1..100] | IsPrime(a) where a is n^2+n+1 ]; // Wesley Ivan Hurt, Jun 16 2014

select(isprime, [j^2+j+1$j=1..200])[];  # Alois P. Heinz, Apr 20 2022

Select[Table[n^2+n+1, {n,250}], PrimeQ] (* Harvey P. Dale, Mar 23 2012 *)

list(lim)=select(n->isprime(n),vector((sqrt(4*lim-3)-1)\2,k,k^2+k+1)) \\ Charles R Greathouse IV, Jul 25 2011

from sympy import isprime
print(list(filter(isprime, (n**2 + n + 1 for n in range(150))))) # Michael S. Branicky, Apr 20 2022

a(n) = A002384(n)^2 + A002384(n) + 1 = (A088503(n-1)^2 + 3)/4 = (A110284(n) + 3)/4. - Ray Chandler, Sep 07 2005

Extended by Ray Chandler, Sep 07 2005

A343774 Primes of the form (c^k+1)/(c+1) not having a representation in the form (b^q-1)/(b-1), where b, c > 1 and k, q > 2.

Original entry on oeis.org

3, 11, 61, 521, 547, 683, 2731, 9091, 13421, 19141, 43691, 61681, 152381, 174763, 185641, 224071, 398581, 909091, 1151041, 1623931, 1824841, 2031671, 2796203, 3341101, 4778021, 5200081, 7027567, 8987221, 10678711, 15790321, 22796593, 25058741, 31224301, 32222107

Offset: 1

Bernard Schott, Apr 29 2021

The exponents k, q are necessarily primes.
Equivalently: primes of the form (c^k+1)/(c+1) that are not Brazilian: intersection of A059055 and A220627.
Except for 3 where k = 3, all the terms of this sequence are of the form (c^k+1)/(c+1) with k prime >= 5.
The only known prime of this form with k prime >= 5 that is not present is 43 = (2^7+1)/(2+1) because also 43 = (7^3+1)/(7+1) = (6^3-1)/(6-1) = 111_6, so 43 belongs to A002383.

			3 = (2^3+1)/(2+1) is not Brazilian, hence 3 is a term.
11 = (2^5+1)/(2+1) is not Brazilian, hence 11 is a term.
547 = (3^7+1)/(3+1) is not Brazilian, hence 547 is a term.
9091 = (10^5+1)/(10+1) is not Brazilian, hence 9091 is a term.

H. Dubner and T. Granlund, Primes of the form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.

Primes of the form (b^k-1)/(b-1) = A085104 (Brazilian primes).
Primes of the form (c^q+1)/(c+1) = A059055.
Primes of the form (b^k-1)/(b-1) and (c^q+1)/(c+1): A002383 \ {3} is a subsequence, but, maybe the intersection (conjecture).
Primes of the form (b^k-1)/(b-1) but not (c^q+1)/(c+1) = A225148.
Primes of the form (c^q+1)/(c+1) but not (b^k-1)/(b-1) = this sequence.
Primes neither of the form (c^q+1)/(c+1) nor (b^k-1)/(b-1) = A343775.

isc(p) = for (b=2, p, my(k=3); while ((x=(b^k+1)/(b+1)) <= p, if (x == p, return (1)); k = nextprime(k+1); ); );
isnotb(p) = for (b=2, p-1, my(d=digits(p, b), md=vecmin(d)); if ((#d > 2) && (md == 1) && (vecmax(d) == 1), return (0)); ); return (1);
isok(p) = isprime(p) && isc(p) && isnotb(p); \\ Michel Marcus, May 01 2021

More terms from Michel Marcus, Apr 30 2021

A343775 Primes that are neither of the form (c^q+1)/(c+1) and nor of the form (b^k-1)/(b-1) for any b, c > 1 and k, q primes > 2.

Original entry on oeis.org

2, 5, 17, 19, 23, 29, 37, 41, 47, 53, 59, 67, 71, 79, 83, 89, 97, 101, 103, 107, 109, 113, 131, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 223, 227, 229, 233, 239, 251, 257, 263, 269, 271, 277, 281, 283, 293, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367

Offset: 1

Bernard Schott, Apr 29 2021

Equivalently, non-Brazilian primes that are not of the form (c^q+1)/(c+1) for some c > 1, q prime > 2.

Equals A220627 \ A059055.

Primes of the form (b^k-1)/(b-1) = A085104 (Brazilian primes).
Primes of the form (c^q+1)/(c+1) = A059055.
Primes of the form (b^k-1)/(b-1) and also (c^q+1)/(c+1): A002383 \ {3} is a subsequence, but, maybe the intersection (conjecture).
Primes of the form (b^k-1)/(b-1) but not (c^q+1)/(c+1) = A225148.
Primes of the form (c^q+1)/(c+1) but not (b^k-1)/(b-1) = A343774.
Primes neither of the form (c^q+1)/(c+1) nor (b^k-1)/(b-1) = this sequence.

isc(p) = for (b=2, p, my(k=3); while ((x=(b^k+1)/(b+1)) <= p, if (x == p, return (1)); k = nextprime(k+1); ); );
isnotb(p) = for (b=2, p-1, my(d=digits(p, b), md=vecmin(d)); if ((#d > 2) && (md == 1) && (vecmax(d) == 1), return (0)); ); return (1);
isok(p) = isprime(p) && !isc(p) && isnotb(p); \\ Michel Marcus, May 01 2021

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).

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A002383 Primes of form k^2 + k + 1.

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A343774 Primes of the form (c^k+1)/(c+1) not having a representation in the form (b^q-1)/(b-1), where b, c > 1 and k, q > 2.

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A343775 Primes that are neither of the form (c^q+1)/(c+1) and nor of the form (b^k-1)/(b-1) for any b, c > 1 and k, q primes > 2.

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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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Author

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Links

Crossrefs

Programs

PARI

PARI

Formula