%I #42 Jul 31 2017 03:22:51

%S 43,73,157,211,241,421,463,601,757,1123,1483,2551,2971,3307,3907,4423,

%T 4831,5701,6007,6163,6481,8191,9901,11131,12211,12433,13807,14281,

%U 19183,20023,20593,21757,22621,22651,23563,24181,26083,26407,27061,28393,31153,35533

%N Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1 where n is not prime.

%C These numbers are Brazilian primes belonging to A085104.

%C Brazilian primes with n prime are A023195, except 3 which is not Brazilian.

%C A085104 = This sequence Union { A023195 \ number 3 }.

%C k + 1 is necessarily prime, but that's not sufficient: 1 + 10 + 100 = 111.

%C Most of these terms come from A185632 which are prime numbers 111_n with n no prime, the first other term is 22621 = 11111_12, the next one is 245411 = 11111_22.

%C Number of terms < 10^k: 0, 2, 9, 23, 64, 171, 477, 1310, 3573, 10098, ..., . - _Robert G. Wilson v_, Apr 15 2017

%H Chai Wah Wu, <a href="/A285017/b285017.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1310 from Robert G. Wilson v)

%H Bernard Schott, <a href="/A125134/a125134.pdf">Les nombres brésiliens</a>, Reprinted from Quadrature, no. 76, April-June 2010, pages 30-38, included here with permission from the editors of Quadrature.

%e 157 = 12^2 + 12 + 1 = 111_12 is prime and 12 is composite.

%p N:= 40000: # to get all terms <= N

%p res:= NULL:

%p for k from 2 to ilog2(N) do if isprime(k) then

%p for n from 2 do

%p p:= (n^(k+1)-1)/(n-1);

%p if p > N then break fi;

%p if isprime(p) and not isprime(n) then res:= res, p fi

%p od fi od:

%p res:= {res}:

%p sort(convert(res,list)); # _Robert Israel_, Apr 14 2017

%t mx = 36000; g[n_] := Select[Drop[Accumulate@Table[n^ex, {ex, 0, Log[n, mx]}], 2], PrimeQ]; k = 1; lst = {}; While[k < Sqrt@mx, If[CompositeQ@k, AppendTo[lst, g@k]; lst = Sort@Flatten@lst]; k++]; lst (* _Robert G. Wilson v_, Apr 15 2017 *)

%o (PARI) isok(n) = {if (isprime(n), forcomposite(b=2, n, d = digits(n, b); if ((#d > 2) && (vecmin(d) == 1) && (vecmax(d)== 1), return(1)););); return(0);} \\ _Michel Marcus_, Apr 09 2017

%o (PARI) A285017_vec(n)={my(h=vector(n,i,1),y,c,z=4,L:list);L=List();forprime(x=3,,forcomposite(m=z,x-1,y=digits(x,m);if((y==h[1..#y])&&2<#y,listput(L,x);z=m;if(c++==n,return(Vec(L))))))} \\ _R. J. Cano_, Apr 18 2017

%Y Cf. A002383, A023195, A053183, A053696, A085104, A088548, A088550, A185632, A190527, A194257.

%K nonn

%O 1,1

%A _Bernard Schott_, Apr 08 2017