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A285017 Primes of the form  1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1 where n is not prime. 7
43, 73, 157, 211, 241, 421, 463, 601, 757, 1123, 1483, 2551, 2971, 3307, 3907, 4423, 4831, 5701, 6007, 6163, 6481, 8191, 9901, 11131, 12211, 12433, 13807, 14281, 19183, 20023, 20593, 21757, 22621, 22651, 23563, 24181, 26083, 26407, 27061, 28393, 31153, 35533 (list; graph; refs; listen; history; text; internal format)
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.

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

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

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

Sequence in context: A144975 A087699 A054807 * A139932 A290635 A176925

Adjacent sequences:  A285014 A285015 A285016 * A285018 A285019 A285020

KEYWORD

nonn

AUTHOR

Bernard Schott, Apr 08 2017

STATUS

approved

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Last modified April 15 08:31 EDT 2021. Contains 342977 sequences. (Running on oeis4.)