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%I #146 Aug 24 2024 18:26:22
%S 7,13,31,43,73,127,157,211,241,307,421,463,601,757,1093,1123,1483,
%T 1723,2551,2801,2971,3307,3541,3907,4423,4831,5113,5701,6007,6163,
%U 6481,8011,8191,9901,10303,11131,12211,12433,13807,14281,17293,19183,19531,20023
%N Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1.
%C 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
%C From _Bernard Schott_, Dec 18 2012: (Start)
%C Also known as Brazilian primes. The primes that are not Brazilian primes are in A220627.
%C 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.
%C 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)
%C It is not known whether there are infinitely many Brazilian primes. See A002383. - _Bernard Schott_, Jan 11 2013
%C Primes of the form (n^p - 1)/(n - 1), where p is odd prime and n > 1. - _Thomas Ordowski_, Apr 25 2013
%C 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
%C From _Bernard Schott_, Apr 08 2017: (Start)
%C Brazilian primes fall into two classes:
%C 1) when n is prime, we get sequence A023195 except 3 which is not Brazilian,
%C 2) when n is composite, we get sequence A285017. (End)
%C 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
%D Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, page 174.
%H Jon E. Schoenfield, <a href="/A085104/b085104.txt">Table of n, a(n) for n = 1..10831</a> (terms up to 10^10; terms 1..3880 from T. D. Noe)
%H Bernard Schott, <a href="/A125134/a125134.pdf">Les nombres brésiliens</a>, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
%F A010051(a(n)) * A088323(a(n)) > 1. - _Reinhard Zumkeller_, Jan 22 2014
%e 13 is a term since it is prime and 13 = 1 + 3 + 3^2 = 111_3.
%e 31 is a term since it is prime and 31 = 1 + 2 + 2^2 + 2^3 + 2^4 = 11111_2.
%e From _Hartmut F. W. Hoft_, May 08 2017: (Start)
%e 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.
%e 2 4 6 10 12 16
%e 2 7 31 127 - 8191 131071
%e 3 13 - 1093 - 797161 -
%e 4 - - - - - -
%e 5 31 - 19531 12207031 305175781 -
%e 6 43 - 55987 - - -
%e 7 - 2801 - - 16148168401 -
%e 8 73 - - - - -
%e 9 - - - - - -
%e 10 - - - - - -
%e 11 - - - - - 50544702849929377
%e 12 157 22621 - - - -
%e 13 - 30941 5229043 - - -
%e 14 211 - 8108731 - - -
%e 15 241 - - - - -
%e 16 - - - - - -
%e 17 307 88741 25646167 2141993519227 - -
%e 18 - - - - - -
%e 19 - - - - - -
%e 20 421 - - 10778947368421 - 689852631578947368421
%e 21 463 - - 17513875027111 - 1502097124754084594737
%e 22 - 245411 - - - -
%e 23 - 292561 - - - -
%e 24 601 346201 - - - -
%e Except for the initial values in the respective sequences the rows and columns as labeled in the matrix are:
%e column 2: A002383 row 2: A000668
%e column 4: A088548 row 3: A076481
%e column 6: A088550 row 4: -
%e column 10: A162861 row 5: A086122.
%e (End)
%t 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 *)
%t 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 *)
%o (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
%o (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
%o (Haskell)
%o a085104 n = a085104_list !! (n-1)
%o a085104_list = filter ((> 1) . a088323) a000040_list
%o -- _Reinhard Zumkeller_, Jan 22 2014
%Y Cf. A189891 (complement), A125134 (Brazilian numbers), A220627 (Primes that are non-Brazilian).
%Y Cf. A003424 (n restricted to prime powers).
%Y Cf. A053696, A086930, A059055.
%Y Equals A023195 \3 Union A285017 with empty intersection.
%Y 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.
%Y Primes of the form (b^k-1)/(b-1) for k=3: A002383, k=5: A088548, k=7: A088550, k=11: A162861.
%K nonn,base
%O 1,1
%A _Amarnath Murthy_ and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 03 2003
%E More terms from _David Wasserman_, Jan 26 2005
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