A085104 -id:A085104 - cp's OEIS frontend

A185632 Primes of the form n^2 + n + 1 where n is nonprime.

3, 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, 22651, 23563, 24181, 26083, 26407, 27061, 28393, 31153, 35533

Offset: 1

Bernard Schott, Dec 18 2012

nonn

These are the primes associated with A182253.
All these numbers are in A002383 but not in A053183.
All the numbers n^2 + n + 1 = 111_n with n >= 2 are by definition Brazilian numbers: A125134. See Links: "Les nombres brésiliens" - Section V.5 page 35.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38.
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, avril-juin 2010, pages 30-38, included here with permission from the editors of Quadrature.

Cf. A002383, A002384, A053182, A053183, A182253.

Cf. A085104, A125134.

Select[Table[If[PrimeQ[n],Nothing,n^2+n+1],{n,200}],PrimeQ] (* Harvey P. Dale, Apr 02 2023 *)

lista(nn) = {for (n = 1, nn, if (! isprime(n) && isprime(p = n^2 + n + 1), print1(p, ", ");););} \\ Michel Marcus, Sep 04 2013

A190300 Composite numbers that are not Brazilian.

Original entry on oeis.org

4, 6, 9, 25, 49, 169, 289, 361, 529, 841, 961, 1369, 1681, 1849, 2209, 2809, 3481, 3721, 4489, 5041, 5329, 6241, 6889, 7921, 9409, 10201, 10609, 11449, 11881, 12769, 16129, 17161, 18769, 19321, 22201, 22801, 24649, 26569, 27889, 29929, 32041, 32761, 36481, 37249, 38809, 39601, 44521, 49729

Offset: 1

N. J. A. Sloane, May 14 2011

nonn

Other than the term 6 and the missing term 121, is this sequence the same as A001248? - Nathaniel Johnston, May 24 2011
From Bernard Schott, Dec 04 2012: (Start)
Yes, because
1) 4 is not a Brazilian number [4 = 100_2].
2) 6 is not a Brazilian number [6 = 110_2 = 20_3 = 12_4].
3) Theorem 1, page 32 of Quadrature article mentioned in links: If n > 7 is not Brazilian, then n is a prime or the square of a prime.
4) Theorem 5, page 37 of Quadrature article mentioned in links: The only square of prime number which is Brazilian is 121 = 11^2 = 11111_3.
(End)
There is an infinity of composite numbers that are not Brazilian: Corollary 2, page 37 of Quadrature article in links (consider the sequence of squares of prime numbers for p >= 13). - Bernard Schott, Dec 17 2012
Also semiprimes that are not Brazilian. - Bernard Schott, Apr 11 2019

			a(10) = p_10^2 = 29^2 = 841.

Robert Israel, Table of n, a(n) for n = 1..10000 (first 208 terms from Robert G. Wilson v)
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A085104, A125134, A189891, A220571, A307507.
Intersection of A002808 and A220570.
Intersection of A001358 and A220570.

4, 6, 9, 25, 49,seq(ithprime(i)^2, i=6..100); # Robert Israel, Apr 17 2019

brazBases[n_] := Select[Range[2, n - 2], Length[Union[IntegerDigits[n, #]]] == 1 &]; Select[Range[2, 10000], ! PrimeQ[#] && brazBases[#] == {} &] (* T. D. Noe, Dec 26 2012 *)
f[n_] := Block[{b = 2}, While[ Length@ Union@ IntegerDigits[n, b] != 1, b++]; b]; k = 4; lst = {}; While[k < 50001, If[ !PrimeQ@ k && 1 + f@ k == k, AppendTo[lst, k]]; k++]; lst (* Robert G. Wilson v, Mar 30 2014 *)

isnotb(n) = my(c=0, d); for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), c=n; break); c++); (c==max(n-3, 0)); \\ A220570
lista(nn) = forcomposite(n=1, nn, if (isnotb(n), print1(n, ", "))); \\ Michel Marcus, Apr 14 2019

a(1) = 2^2 = p_1^2, a(2) = 2*3 = p_1*p_2, a(3) = 3^2 = p_2^2, a(4) = 5^2 = p_3^2, a(5) = 7^2 = p_4^2, a(6) = 13^2 = p_6^2, ..., for n >= 6, a(n) = p_n^2, where p_k is the k-th prime number. - Bernard Schott, Dec 04 2012

a(6)-a(24) from Nathaniel Johnston, May 24 2011

a(25) onward from Robert G. Wilson v, Mar 30 2014

A190527 Primes of the form p^4 + p^3 + p^2 + p + 1, where p is prime.

Original entry on oeis.org

31, 2801, 30941, 88741, 292561, 732541, 3500201, 28792661, 39449441, 48037081, 262209281, 1394714501, 2666986681, 3276517921, 4802611441, 5908670381, 12936304421, 16656709681, 19408913261, 24903325661, 37226181521, 43713558101, 52753304641, 64141071121, 96427561501, 100648118041

Offset: 1

Bernard Schott, Dec 20 2012

nonn

These primes are generated by exactly A065509, cf. 2nd formula.
These numbers are repunit primes 11111_p, so they are Brazilian primes (A085104).
When p^4 + p^3 + p^2 + p + 1 = sigma(p^4) is prime, then it equals A193574(p^4), so that this sequence is a subsequence of A193574; by definition it is also a subsequence of A053699 and A131992. - Hartmut F. W. Hoft, May 05 2017

			a(3) = 30941 = 11111_13 = 13^4 + 13^3 + 13^2 + 13^1 + 1 is prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1100

Cf. A049409 (n^4 + ... + 1 is prime), A065509 (primes among these n), A193574.
Subsequence of A088548 (primes n^4 + ... + 1) and A085104 ("Brazilian" primes, of the form 1 + n + n^2 + ... + n^k).
Intersection of A000040 (primes) and A131992 (p^4 + ... + 1), subsequence of A053699 (n^4 + ... + 1).

[p: p in PrimesUpTo(600) | IsPrime(p) where p is p^4 +p^3+p^2+p+1]; // Vincenzo Librandi, May 06 2017

a190527[n_] := Select[Map[(Prime[#]^5-1)/(Prime[#]-1)&, Range[n]], PrimeQ]
a190527[100] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)
Select[#^4 + #^3 + #^2 + # + 1 &/@Prime[Range[100]], PrimeQ] (* Vincenzo Librandi, May 06 2017 *)

[q|p<-primes(100),ispseudoprime(q=(p^5-1)\(p-1))]
A190527_vec(N)=[(p^5-1)\(p-1)|p<-A065509_vec(N)] \\ M. F. Hasler, Mar 03 2020

a(n) = A193574(A065509(n)^4). - Hartmut F. W. Hoft, May 08 2017

a(n) = A053699(A065509(n)) = A000203(A065509(n)^4). - M. F. Hasler, Mar 03 2020

a(7) corrected and a(18)-a(26) added by Hartmut F. W. Hoft, May 05 2017

Edited by M. F. Hasler, Mar 06 2020

A257521 Odd Brazilian numbers.

Original entry on oeis.org

7, 13, 15, 21, 27, 31, 33, 35, 39, 43, 45, 51, 55, 57, 63, 65, 69, 73, 75, 77, 81, 85, 87, 91, 93, 95, 99, 105, 111, 115, 117, 119, 121, 123, 125, 127, 129, 133, 135, 141, 143, 145, 147, 153, 155, 157, 159, 161, 165, 171, 175, 177, 183, 185, 187, 189, 195

Offset: 1

Daniel Lignon, Apr 27 2015

All even integers 2p >=8 are Brazilian numbers (A125134), because 2p=2(p-1)+2 is written 22 in base p-1 if p-1>2, that is true if p >=4. But, among Brazilian numbers, there are also odd ones...

The only square of a prime is 121. - Robert G. Wilson v, May 21 2015

Daniel Lignon and Robert Israel, Table of n, a(n) for n = 1..10000 (first 703 from Daniel Lignon)

Cf. A125134 (Brazilian numbers), A253261 (odd Brazilian squares).

Cf. A085104 (prime Brazilian numbers).

N:= 1000: # to get all terms <= N
for b from 2 to floor(N/2-1) do
   dk:= 1 + (b mod 2);
   for j from 1 to b-1 by 2 do
     for k from dk by dk do
       if j=1 and k=1 then next fi;
       x:= j*(b^(k+1)-1)/(b-1);
       if x > N then break fi;
       B[x]:= 1;
     od
   od
od:
sort(map(op,[indices(B)])); # Robert Israel, May 27 2015

fQ[n_] := Block[{b = 2}, While[b < n - 1 && Length[ Union[ IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; Select[1 + 2 Range@100, fQ] (* Robert G. Wilson v, May 21 2015 *)

forstep(n=5, 300, 2, for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), print1(n, ", "); break))) \\ Derek Orr, Apr 30 2015

A285017 Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1 where n is not prime.

Original entry on oeis.org

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

Offset: 1

Bernard Schott, Apr 08 2017

nonn

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

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

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.

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

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

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 *)

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

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

A308851 Numbers >= 2 all of whose divisors > 1 are Brazilian.

Original entry on oeis.org

7, 13, 31, 43, 73, 91, 127, 157, 211, 217, 241, 301, 307, 403, 421, 463, 511, 559, 601, 757, 889, 949, 1093, 1099, 1123, 1333, 1477, 1483, 1651, 1687, 1723, 2041, 2149, 2263, 2551, 2743, 2801, 2821, 2947, 2971, 3133, 3139, 3241, 3307, 3541, 3907, 3913, 3937

Offset: 1

Bernard Schott, Jun 28 2019

nonn

The terms of this sequence are the Brazilian primes and the products of two or more distinct Brazilian primes.

There are no even numbers because 2 is not Brazilian.

			91 is a term because all divisors of 91 that are > 1: {7, 13, 91} are Brazilian numbers with 7 = 111_2, 13 = 111_3 and 91 = 77_12.

Amiram Eldar, Table of n, a(n) for n = 1..1000
Index entries for sequences related to Brazilian numbers

Cf. A085104 (subsequence), A125134.

Similar with even numbers: A000079, with odd numbers: A005408, with palindromes: A062687, with repdigits: A190217.

brazQ[n_] := Block[{k, b, ok}, If[FindInstance[k (1 + b) == n && 1 < b < n - 1 && 0 < k < b, {k, b}, Integers] != {}, True, b = 2; ok = False; While[1 + b + b^2 <= n && ! ok, ok = Length@ Union@ IntegerDigits[n, b++] == 1]; ok]]; Select[ Range[3, 4000, 2], AllTrue[ Rest@ Divisors@ #, brazQ] &] (* Giovanni Resta, Jun 29 2019 *)
max = 5000; fQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; A125134 = Select[Range[4, max], fQ]; Select[Range[2, max], Intersection[A125134, Rest[Divisors[#]]] == Rest[Divisors[#]] &] (* Vaclav Kotesovec, Jun 29 2019, using a subroutine from T. D. Noe *)

isb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1)));
isok(n) = {fordiv(n, d, if ((d>1) && ! isb(d), return (0));); return (1);} \\ Michel Marcus, Jun 29 2019

A119959 p^2-p+1 central polygonal numbers with prime indices A002061(prime(n)).

Original entry on oeis.org

3, 7, 21, 43, 111, 157, 273, 343, 507, 813, 931, 1333, 1641, 1807, 2163, 2757, 3423, 3661, 4423, 4971, 5257, 6163, 6807, 7833, 9313, 10101, 10507, 11343, 11773, 12657, 16003, 17031, 18633, 19183, 22053, 22651, 24493, 26407, 27723, 29757, 31863

Offset: 1

Alexander Adamchuk, Aug 02 2006

nonn

Prime terms belong to A074268, which is a subset of A002383, A087126, A034915, A085104.

In every interval of prime(n)^2 integers, a(n) is the number that are not divisible by prime(n) plus the number that are divisible by prime(n)^2. - Peter Munn, Dec 12 2020

Cf. A002061, A074268, A002383, A087126, A034915, A068468, A085104, A083558.

Mathematica
```
Table[Prime[n]^2-Prime[n]+1,{n,1,100}]
```

a(n) = {my(p = prime(n)); p^2 - p + 1; } \\ Amiram Eldar, Nov 07 2022

a(n) = prime(n)^2 - prime(n) + 1.
a(n) = A036689(n)+1. - R. J. Mathar, Aug 13 2019
Product_{n>=1} (1 - 1/a(n)) = zeta(6)/(zeta(2)*zeta(3)) (A068468). - Amiram Eldar, Nov 07 2022

A194257 Primes of the form p^6 + p^5 + p^4 + p^3 + p^2 + p + 1 when p is prime.

Original entry on oeis.org

127, 1093, 19531, 5229043, 25646167, 917087137, 52379047267, 153436090543, 502628805631, 11016462577051, 18871143464293, 251059142817757, 1812169199976451, 1940350890330343

Offset: 1

Bernard Schott, Dec 21 2012

nonn

These primes are generated by exactly A163268.
This sequence is included in A088550.
These numbers are repunit primes 1111111_n, so they are Brazilian primes and are terms of A085104.
Subsequence of A088550. - Hartmut F. W. Hoft, May 05 2017

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A085104, A088550, A100330, A163268.

select(isprime, map(p -> add(p^i,i=0..6), select(isprime, [2,seq(i,i=3..1000,2)]))); # Robert Israel, May 05 2017

a194257[n_] := Select[Map[(Prime[#]^7-1)/(Prime[#]-1)&, Range[n]], PrimeQ]
a194257[70] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)
Select[Table[Total[p^Range[0,6]],{p,Prime[Range[100]]}],PrimeQ] (* Harvey P. Dale, Mar 09 2024 *)

a(n) = A193574(A163268(n)^6). - Hartmut F. W. Hoft, May 08 2017

A088323 Number of numbers b>1 such that n is a repunit in base b representation.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Reinhard Zumkeller, Nov 06 2003

is a(n) < 4 ?;
n>2: a(n) > 0 as n = (n-1)^1 + (n-1)^0.
a(A119598(n)) > 3; a(A053696(n)) > 2; a(A085104(n)) > 2. - Reinhard Zumkeller, Jan 22 2014

			a(31)=3: 31 = 2^4+2^3+2^2+2^1+2^0 = 5^2+5^1+5^0 = 30^1+30^0.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
Eric Weisstein's World of Mathematics, Repunit

Cf. A000225, A003462, A002275, A068953.

a088323 n = sum $ map (f n) [2 .. n-1] where
   f x b = if x == 0 then 1 else if d /= 1 then 0 else f x' b
                                 where (x',d) = divMod x b
-- Reinhard Zumkeller, Jan 22 2014

Example corrected by Reinhard Zumkeller, Jan 22 2014

A165210 Primes of the form (6^m - 1)/5.

Original entry on oeis.org

7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371

Offset: 1

Rick L. Shepherd, Sep 07 2009

nonn

Prime repunits in base 6 whose representation consists of m 1's. The exponents m are in A004062. a(5) and a(6) have 55 and 99 decimal digits, respectively.

			a(2) = (6^A004062(2) - 1)/5 = (6^3 - 1)/5 = 215/5 = 43, which is 111_6.

Vincenzo Librandi, Table of n, a(n) for n = 1..9

Cf. A004062, A003464, A085104, A000668, A076481, A086122, A102170, A004022.

[a: n in [1..200] | IsPrime(a) where a is  (6^n-1) div 5 ]; // Vincenzo Librandi, Dec 09 2011

Select[Table[(6^n-1)/5, {n,0,2000}], PrimeQ] (* Vincenzo Librandi, Dec 09 2011 *)

a(n) = (6^A004062(n) - 1)/5.

cp's OEIS Frontend

A185632 Primes of the form n^2 + n + 1 where n is nonprime.

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A190300 Composite numbers that are not Brazilian.

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A190527 Primes of the form p^4 + p^3 + p^2 + p + 1, where p is prime.

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A257521 Odd Brazilian numbers.

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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.

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A308851 Numbers >= 2 all of whose divisors > 1 are Brazilian.

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A119959 p^2-p+1 central polygonal numbers with prime indices A002061(prime(n)).

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