OFFSET
1,1
COMMENTS
These numbers are Brazilian primes belonging to A085104.
Brazilian primes with n prime are A023195, except 3 which is not Brazilian.
k + 1 is necessarily prime, but that's not sufficient: 1 + 10 + 100 = 111.
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.
Number of terms < 10^k: 0, 2, 9, 23, 64, 171, 477, 1310, 3573, 10098, ..., . - Robert G. Wilson v, Apr 15 2017
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10000 (terms 1..1310 from Robert G. Wilson v)
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, April-June 2010, pages 30-38, included here with permission from the editors of Quadrature.
EXAMPLE
157 = 12^2 + 12 + 1 = 111_12 is prime and 12 is composite.
MAPLE
N:= 40000: # to get all terms <= N
res:= NULL:
for k from 2 to ilog2(N) do if isprime(k) then
for n from 2 do
p:= (n^(k+1)-1)/(n-1);
if p > N then break fi;
if isprime(p) and not isprime(n) then res:= res, p fi
od fi od:
res:= {res}:
sort(convert(res, list)); # Robert Israel, Apr 14 2017
MATHEMATICA
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 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Apr 08 2017
STATUS
approved