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A085104
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Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1.
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65
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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
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OFFSET
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1,1
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COMMENTS
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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
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
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REFERENCES
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Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, page 174.
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LINKS
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Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
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FORMULA
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EXAMPLE
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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.
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:
(End)
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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Cf. A003424 (n restricted to prime powers).
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KEYWORD
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nonn,base,changed
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AUTHOR
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Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 03 2003
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EXTENSIONS
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STATUS
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approved
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