A085104 Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1.

7, 13, 31, 43, 73, 127, 157, 211, 241, 307, 421, 463, 601, 757, 1093, 1123, 1483, 1723, 2551, 2801, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9901, 10303, 11131, 12211, 12433, 13807, 14281, 17293, 19183, 19531, 20023

Offset: 1

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 03 2003

Keywords

• nonn
• base

Comments

Primes that are base-b repunits with three or more digits for at least one b >= 2: Primes in A053696. Subsequence of A000668 U A076481 U A086122 U A165210 U A102170 U A004022 U ... (for each possible b). - Rick L. Shepherd, Sep 07 2009

From Bernard Schott, Dec 18 2012: (Start)

Also known as Brazilian primes. The primes that are not Brazilian primes are in A220627.

The number of terms k+1 is always an odd prime, but this is not enough to guarantee a prime, for example 111 = 1 + 10 + 100 = 3*37.

The inverses of the Brazilian primes form a convergent series; the sum is slightly larger than 0.33 (see Theorem 4 of Quadrature article in the Links). (End)

It is not known whether there are infinitely many Brazilian primes. See A002383. - Bernard Schott, Jan 11 2013

Primes of the form (n^p - 1)/(n - 1), where p is odd prime and n > 1. - Thomas Ordowski, Apr 25 2013

Number of terms less than 10^n: 1, 5, 14, 34, 83, 205, 542, 1445, 3880, 10831, 30699, 88285, ..., . - Robert G. Wilson v, Mar 31 2014

From Bernard Schott, Apr 08 2017: (Start)

Brazilian primes fall into two classes:

1) when n is prime, we get sequence A023195 except 3 which is not Brazilian,

2) when n is composite, we get sequence A285017. (End)

The conjecture proposed in Quadrature "No Sophie Germain prime is Brazilian (prime)" (see link Bernard Schott, Quadrature, Conjecture 1, page 36) is false. Thanks to Giovanni Resta, who found that a(856) = 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)73 is the 141385th Sophie Germain prime. - _Bernard Schott, Mar 08 2019

Examples

			13 is a term since it is prime and 13 = 1 + 3 + 3^2 = 111_3.
31 is a term since it is prime and 31 = 1 + 2 + 2^2 + 2^3 + 2^4 = 11111_2.
From _Hartmut F. W. Hoft_, May 08 2017: (Start)
The sequence represented as a sparse matrix with the k-th column indexed by A006093(k+1), primes minus 1, and row n by A000027(n+1). Traversing the matrix by counterdiagonals produces a non-monotone ordering.
    2    4      6        10             12          16
2  7    31     127      -              8191        131071
3  13   -      1093     -              797161      -
4  -    -      -        -              -           -
5  31   -      19531    12207031       305175781   -
6  43   -      55987    -              -           -
7  -    2801   -        -              16148168401 -
8  73   -      -        -              -           -
9  -    -      -        -              -           -
10  -    -      -        -              -           -
11  -    -      -        -              -           50544702849929377
12  157  22621  -        -              -           -
13  -    30941  5229043  -              -           -
14  211  -      8108731  -              -           -
15  241  -      -        -              -           -
16 -    -      -        -              -           -
17  307  88741  25646167 2141993519227  -           -
18  -    -      -        -              -           -
19  -    -      -        -              -           -
20  421  -      -        10778947368421 -           689852631578947368421
21  463  -      -        17513875027111 -           1502097124754084594737
22  -    245411 -        -              -           -
23  -    292561 -        -              -           -
24  601  346201 -        -              -           -
Except for the initial values in the respective sequences the rows and columns as labeled in the matrix are:
column  2:  A002383            row 2:  A000668
column  4:  A088548            row 3:  A076481
column  6:  A088550            row 4:  -
column 10:  A162861            row 5:  A086122.
(End)
		

References

• Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, page 174.

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..10831 (terms up to 10^10; terms 1..3880 from T. D. Noe)
• Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Crossrefs

Cf. A189891 (complement), A125134 (Brazilian numbers), A220627 (Primes that are non-Brazilian).

Cf. A003424 (n restricted to prime powers).

Cf. A053696, A086930, A059055.

Equals A023195 \3 Union A285017 with empty intersection.

Primes of the form (b^k-1)/(b-1) for b=2: A000668, b=3: A076481, b=5: A086122, b=6: A165210, b=7: A102170, b=10: A004022.

Primes of the form (b^k-1)/(b-1) for k=3: A002383, k=5: A088548, k=7: A088550, k=11: A162861.

Programs

Haskell

a085104 n = a085104_list !! (n-1)
a085104_list = filter ((> 1) . a088323) a000040_list
-- Reinhard Zumkeller, Jan 22 2014

Mathematica

max = 140; maxdata = (1 - max^3)/(1 - max); a = {}; Do[i = 1; While[i = i + 2; cc = (1 - m^i)/(1 - m); cc <= maxdata, If[PrimeQ[cc], a = Append[a, cc]]], {m, 2, max}]; Union[a] (* Lei Zhou, Feb 08 2012 *)
f[n_] := Block[{i = 1, d, p = Prime@ n}, d = Rest@ Divisors[p - 1]; While[ id = IntegerDigits[p, d[[i]]]; id != Reverse@ id || Union@ id != {1}, i++]; d[[i]]]; Select[ Range[2, 60], 1 + f@# != Prime@# &] (* Robert G. Wilson v, Mar 31 2014 *)

PARI

list(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))));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jan 08 2013

PARI

A085104_vec(N,L=List())=forprime(K=3,logint(N+1,2),for(n=2,sqrtnint(N-1,K-1),isprime((n^K-1)\(n-1))&&listput(L,(n^K-1)\(n-1))));Set(L) \\ M. F. Hasler, Jun 26 2018

Formula

A010051(a(n)) * A088323(a(n)) > 1. - Reinhard Zumkeller, Jan 22 2014

Extensions

More terms from David Wasserman, Jan 26 2005