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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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

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

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.

FORMULA

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

EXAMPLE

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)

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 *)

PROG

(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

(Haskell)

a085104 n = a085104_list !! (n-1)

a085104_list = filter ((> 1) . a088323) a000040_list

-- Reinhard Zumkeller, Jan 22 2014

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.

Sequence in context: A240680 A308851 A298027 * A162652 A306889 A181141