A085104 -id:A085104 - cp's OEIS frontend

A193366 Primes of the form n^4 + n^3 + n^2 + n + 1 where n is nonprime.

5, 22621, 245411, 346201, 637421, 837931, 2625641, 3835261, 6377551, 15018571, 16007041, 21700501, 30397351, 35615581, 52822061, 78914411, 97039801, 147753211, 189004141, 195534851, 209102521, 223364311, 279086341, 324842131, 421106401, 445120421, 566124791, 693025471, 727832821, 745720141, 880331261, 943280801, 987082981, 1544755411, 1740422941

Offset: 1

Jonathan Vos Post, Dec 20 2012

Note that there are no primes of the form n^3 + n^2 + n + 1 = (n+1)*(n^2+1) as irreducible components over Z.
From Bernard Schott, May 15 2017: (Start)
These are the primes associated with A286094.
A088548 = A190527 Union {This sequence}.
All the numbers of this sequence n^4 + n^3 + n^2 + n + 1 = 11111_n with n > 1 are Brazilian numbers, so belong to A125134 and A085104. (End)

			a(1) = 1^4 + 1^3 + 1^2 + 1 + 1 = 5.
a(2) = 12^4 + 12^3 + 12^2 + 12 + 1 = 22621.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, avril-juin 2010, pages 30-38.

Subsequence of A088548.
Cf. A000040, A018252, A185632, A192321.
Cf. A049409, A053699, A065509, A085104, A088548, A125134, A190527, A286094.

for n from 1 to 150 do p(n):= 1+n+n^2+n^3+n^4;
if tau(n)>2 and isprime(p(n)) then print(n,p(n)) else fi od: # Bernard Schott, May 15 2017

Select[Map[Total[#^Range[0, 4]] &, Select[Range@ 204, ! PrimeQ@ # &]], PrimeQ] (* Michael De Vlieger, May 15 2017 *)

print1(5);forcomposite(n=4,1e3,if(isprime(t=n^4+n^3+n^2+n+1),print1(", "t))) \\ Charles R Greathouse IV, Mar 25 2013

{n^4 + n^3 + n^2 + n + 1 where n is in A018252}.

A285642 Smallest Brazilian prime in base n, or 0 if no such prime exists.

Original entry on oeis.org

7, 13, 0, 31, 43, 2801, 73, 0, 1111111111111111111, 50544702849929377, 157, 30941, 211, 241, 0, 307

Offset: 2

Bernard Schott, Apr 23 2017

nonn

Also the smallest prime of the form (n^k - 1)/(n - 1) with k > 2. The corresponding values of k are in A128164.
For n = 18, a(n) = (18^25667 - 1)/17 as explained in the extension of A128164, but it is too large to write in the Data field.
Differs from A084738: in A084738, the primes of the form (n^2 - 1)/(n - 1) = n + 1 are included, for instance 7 = 6 + 1 = 11_6 but not included here, so a(6) = 43 = 111_6.
As mentioned by Dubner, see link, when n is a power of a prime ( >= 2 ), the number (n^k - 1)/(n - 1) with k > 2 is usually composite, so a(4) = a(9) = a(16) = a(25) = 0 for instance, exception a(8) = 73.
Values of a(19)-a(31): {109912203092239643840221, 421, 463, 245411, 292561, 601, 0, 321272407, 757, 637421, 732541, 837931, 917087137}. - Michael De Vlieger, Apr 24 2017

			a(7) = (7^5 - 1)/6 = 11111_7 =  1 + 7 + 7^2 + 7^3 + 7^4 = 2801.
a(10) is the repunit R_19 which is a string of nineteen 1's.

H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.

Cf. A084738, A084740, A085104, A128164.

Table[Function[m, If[m > 0, k = 3; While[! PrimeQ[Set[x, (m^k - 1)/(m - 1)]], k++]; x, 0]]@ If[Set[e, GCD @@ #[[All, -1]]] > 1, {#, IntegerExponent[n, #]} &@ Power[n, 1/e] /. {{k_, m_} /; Or[Not[PrimePowerQ@ m], Prime@ m, FactorInteger[m][[1, 1]] == 2] :> 0, {k_, m_} /; m > 1 :> n}, n] &@ FactorInteger@ n, {n, 2, 17}] (* Michael De Vlieger, Apr 24 2017 *)

A286094 Nonprime numbers n such that n^4 + n^3 + n^2 + n + 1 is prime.

Original entry on oeis.org

1, 12, 22, 24, 28, 30, 40, 44, 50, 62, 63, 68, 74, 77, 85, 94, 99, 110, 117, 118, 120, 122, 129, 134, 143, 145, 154, 162, 164, 165, 172, 175

Offset: 1

Bernard Schott, May 02 2017

nonn

A065509 Union {this sequence} = A049409.

The corresponding prime numbers n^4 + n^3 + n^2 + n + 1 = 11111_n are in A193366; these Brazilian primes, except 5 which is not Brazilian, belong to A085104 and A285017.

			12 is in the sequence because 12^4 + 12^3 + 12^2 + 12 + 1 = 11111_12 = 22621, which is prime.

R. J. Cano, Table of n, a(n) for n = 1..10000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38.

Cf. A049409, A065509, A085104, A088548, A125134, A190527, A193366, A285017.

Select[Range@ 414, And[! PrimeQ@ #, PrimeQ[Total[#^Range[0, 4]]]] &] (* Michael De Vlieger, May 03 2017 *)

isok(n)=if(n==1,5,if(ispseudoprime(n), 0, isprime(fromdigits([1, 1, 1, 1, 1], n))));
getfirstterms(n)={my(L:list, c:small); L=List(); c=0; forstep(k=1, +oo, 1, if(isok(k), listput(L, k); if(c++==n, break))); return(Vec(L))} \\ R. J. Cano, May 09 2017

A306845 Sophie Germain primes which are Brazilian.

Original entry on oeis.org

28792661, 78914411, 943280801, 7294932341, 30601685951, 919548423641, 2275869057821, 4172851565741, 4801096143881, 27947620155401, 29586967653101, 43573806645461, 119637719305001, 124484682222941, 148908227169101, 172723673300501

Offset: 1

Bernard Schott, Mar 13 2019

nonn

These terms point out that the conjecture proposed in Quadrature "No Sophie Germain prime is Brazilian (prime)" (see link) was false.
Giovanni Resta has found the first counterexample of Sophie Germain prime which is Brazilian. It's the 141385th Sophie Germain prime 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)73. The other counterexamples have been found by _Michel Marcus.
These numbers are relatively rare: only 25 terms < 10^15.
The 47278 initial terms of this sequence are of the form (11111)_b. The successive bases b are 73, 94, 175, 292, 418, 979, 1228, 1429, ...
The first term which is not of this form has 32 digits, it is 14781835607449391161742645225951 = 1 + 1309 + ... + 1309^9 + 1309^10 = (11111111111)_1309 with a string of eleven 1's. In this case, the successive bases b are 1309, 1348, 2215, 2323, 2461, ...
If (b^q - 1)/(b - 1) is a term, necessarily q (prime) == 5 (mod 6) and b == 1 (mod 3). The smallest term for each pair (q,b) is: (5,73), (11,1309), (17,1945), (23,20413), (29,5023), (41,9565), (47,2764) (See link Jon Grantham, Hester Graves).
Other smallest pairs (q, b) are: (53, 139492), (59, 154501), (71, 7039), (83, 9325), (89, 78028), (101, 8869), (107, 86503), (113, 89986), (131, 429226), (137, 929620), (149, 1954), (167, 175), (173, 1368025). - David A. Corneth, Mar 13 2019

			78914411 is a term because 2 * 78914411 + 1 = 157828823 is prime, so 78914411 is Sophie Germain prime, then, 78914411 = 1 + 94 + 94^2 + 94^3 + 94^4 = (11111)_94 and 78914411 is also a Brazilian prime.

Jon Grantham, Hester Graves, Brazilian Primes Which Are Also Sophie Germain Primes, arXiv:1903.04577 [math.NT], 2019.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Intersection of A005384 and A085104.

Cf. A007528, A306849.

lista(lim)=my(v=List(), t, k); for(n=2, sqrt(lim), t=1+n; k=1; while((t+=n^k++)<=lim, if(isprime(t) && isprime(2*t+1), listput(v, t)))); v = vecsort(Vec(v), , 8); \\ Michel Marcus, Mar 13 2019

A306849 Brazilian primes that are also the lesser of a pair of twin primes.

Original entry on oeis.org

2801, 637421, 2625641, 78914411, 195534851, 7294932341, 19408913261, 57765899591, 133311428141, 212872312241, 1508520377381, 1960226457281, 5412080545901, 11543487851801, 19383356741711, 20748237948131, 24212632812551, 25413171899021, 28240486488581, 46922470889141

Offset: 1

Bernard Schott, Mar 13 2019

As for Sophie Germain primes which are Brazilian (A306845), these terms are relatively rare (only 28 terms < 10^15).
The first 26051 terms of this sequence are of the form (11111)_b. The successive bases b are 7, 28, 40, 94, 118, 292, 373, 490, 604, 679, 1108, 1183, ... These 26051 terms end in 1: If base b ends in 1 or 6, (11111)_b ends in 5 and cannot be prime; if base b ends in another digit, then (11111)_b always ends in 1.
The first term which is not of this form has 31 digits; it's 1425663266336265377189900884061 = 1 + 1036 + ... + 1036^9 + 1036^10 = (11111111111)_1036 with a string of eleven 1's. In this case, the successive bases are 1036, 2089, 6961, 7894, 9775, ...
If (b^q - 1)/(b - 1) is a term, necessarily q (prime) == 5 (mod 6) and b == 1 (mod 3). The smallest term for each pair (q,b) is (5,7), (11,1036), (17,1603), (23,6697), (29,2779), (41,26719), (47,98506), (53,2110).

			2801 is a term because 2801 + 2 = 2803 is prime, so 2801 is a lesser of twin primes, then 2801 = 1 + 7 + 7^2 + 7^3 + 7^4 = (11111)_7 and 2801 is also a Brazilian prime.

Intersection of A001359 and A085104.

Cf. A001097, A006512, A306845, A306889.

lista(lim)=my(v=List(), t, k); for(n=2, sqrt(lim), t=1+n; k=1; while((t+=n^k++)<=lim, if(isprime(t) && isprime(t+2), listput(v, t)))); v = vecsort(Vec(v), , 8); \\ Michel Marcus, Mar 14 2019

Terms computed by Giovanni Resta and Michel Marcus, Mar 13 2019

A306889 Brazilian primes that are also the greater of a pair of twin primes.

Original entry on oeis.org

7, 13, 31, 43, 73, 241, 421, 463, 601, 1093, 1483, 1723, 2551, 2971, 3541, 4423, 8011, 10303, 17293, 19183, 20023, 22621, 23563, 24181, 27061, 31153, 35533, 41413, 42643, 43891, 46441, 47743, 53593, 55933, 60763, 83233, 84391, 95791, 98911, 123553, 143263, 156421, 164431

Offset: 1

Michel Marcus, Mar 15 2019

R. J. Mathar, Table of n, a(n) for n = 1..417

Intersection of A006512 and A085104.

Cf. A306845, A306849.

lista(lim)=my(v=List(), t, k); for(n=2, sqrt(lim), t=1+n; k=1; while((t+=n^k++)<=lim, if(isprime(t) && isprime(t-2), listput(v, t)))); v = vecsort(Vec(v), , 8); \\ Michel Marcus, Mar 15 2019

A340796 a(n) is the smallest number with exactly n divisors that are Brazilian.

Original entry on oeis.org

1, 7, 14, 24, 40, 48, 60, 84, 140, 144, 120, 168, 252, 700, 240, 336, 560, 360, 420, 672, 1120, 2304, 960, 720, 1008, 1080, 840, 2184, 1800, 1260, 2016, 5376, 8960, 2160, 1680, 2880, 4032, 3600, 7056, 19600, 3960, 2520, 3360, 6480, 9072, 9900, 6300, 11520, 16128

Offset: 0

Bernard Schott, Jan 21 2021

Primes can be partitioned into Brazilian primes and non-Brazilian primes. If two distinct primes each larger than 11 are in the same category then the larger one has a multiplicity that is smaller than or equal to that of the smaller prime. - David A. Corneth, Jan 24 2021

			Of the eight divisors of 24, three are Brazilian numbers: 8, 12 and 24, and there is no smaller number with three Brazilian divisors, hence a(3) = 24.

David A. Corneth, Table of n, a(n) for n = 0..427
David A. Corneth, More terms
Wikipédia, Nombre brésilien (in French).
Index entries for sequences related to Brazilian numbers.

Cf. A085104, A125134, A190300, A220570, A308851, A340795, A340797.

Similar with: A087997 (palindromes), A333456 (Niven), A335038 (Zuckerman).

brazQ[n_] := Module[{b = 2, found = False}, While[b < n - 1 && Length[Union[IntegerDigits[n, b]]] > 1, b++]; b < n - 1]; d[n_] := DivisorSum[n, 1 &, brazQ[#] &]; m = 30; s = Table[0, {m}]; c = 0; n = 1; While[c < m, i = d[n]; If[i < m && s[[i + 1]] == 0, c++; s[[i + 1]] = n]; n++]; s (* Amiram Eldar, Jan 21 2021 *)

isokb(n) = for(b=2, n-2, d=digits(n, b); if(vecmin(d)==vecmax(d), return(1))); \\ A125134
isok(k, n) = sumdiv(k, d, isokb(d)) == n;
a(n) = my(k=1); while (!isok(k, n), k++); k; \\ Michel Marcus, Jan 23 2021

More terms from Amiram Eldar, Jan 21 2021

A086930 Smallest b>1 such that in base b representation the n-th prime is a repunit.

Original entry on oeis.org

2, 4, 2, 10, 3, 16, 18, 22, 28, 2, 36, 40, 6, 46, 52, 58, 60, 66, 70, 8, 78, 82, 88, 96, 100, 102, 106, 108, 112, 2, 130, 136, 138, 148, 150, 12, 162, 166, 172, 178, 180, 190, 192, 196, 198, 14, 222, 226, 228, 232, 238, 15, 250, 256, 262, 268, 270, 276, 280, 282

Offset: 2

Reinhard Zumkeller, Sep 21 2003

From Robert G. Wilson v, Mar 26 2014: (Start)
Obviously the first prime number, 2, can never become a repunit since it is even; therefore this sequence has the offset of 2.
Most terms, a(n), are one less than the n-th prime; e.g., for a(8) the eighth prime is 19_10 = 11_18. Therefore a(n) <= Pi(n)-1.
However there are some terms for which a(n) occurs before Pi(n)-1; e.g., for a(14) the fourteenth prime is 43_10 = 111_6.
Those indices, i, are: 4, 6, 11, 14, 21, 31, 37, 47, 53, 63, 82, 90, ..., . Prime(i) = A085104.
In those cases a(n) is a proper divisor of Prime(n)-1.
(End)

			n=6: A000040(6) = 13 = 1*3^2 + 1*3^1 + 1*3^0: ternary(13)='111' and binary(13)='1101', therefore a(6)=3.

Robert G. Wilson v, Table of n, a(n) for n = 2..1001
Eric Weisstein's World of Mathematics, Repunit

Cf. A000225, A003462, A002450, A003463, A003464, A023000, A023001, A002452, A002275, A004022, A016123, A016125, A085104.

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]]]; Array[f, 60, 2]

A187823 Primes of the form (p^x - 1)/(p^y - 1), where p is prime, y > 1, and y is the largest proper divisor of x.

Original entry on oeis.org

5, 17, 73, 257, 757, 65537, 262657, 1772893, 4432676798593, 48551233240513, 378890487846991, 3156404483062657, 17390284913300671, 280343912759041771, 319913861581383373, 487014306953858713, 5559917315850179173, 7824668707707203971, 8443914727229480773, 32564717507686012813

Offset: 1

Bernard Schott, Dec 27 2012

nonn

Complement of A023195 relative to A003424.
Only eight primes of this form don't exceed 1.275*10^10 (see Bateman and Stemmler):
(1) three of the form (p^9 - 1)/(p^3 - 1): 73 (p=2), 757 (p=3), 1772893 (p=11);
(2) four of the form (2^x - 1)/(2^y - 1) with x = 2y: 5 (x=4), 17 (x=8), 257 (x=16), 65537 (x=32); and
(3) the prime 262657 = (2^27 - 1)/(2^9 - 1).
Some of these prime numbers are not Brazilian, these are Fermat primes > 3: 5, 17, 257, 65537, so they are in A220627.
The other primes are Brazilian so they are in A085104, example: (p^9 - 1)/(p^3 - 1) = 111_{p^3} with 73 = 111_8, 757 = 111_27, 1772893 = 111_1331, also 262657 = 111_512 [See section V.4 of Quadrature article in Links] (comment improved in Mar 03 2023).
Comments from Don Reble, Jul 28 2022 (Start)
This is an easy sequence that looks hard.
Note that x must be a power of a prime; otherwise (p^x-1)/(p^y-1) has too many cyclotomic factors.
Almost all values are (p^9-1)/(p^3-1). The exceptions below 10^45
    are the Fermat primes 5, 17, 257, 65537 and also
    262657, 4432676798593, 5559917315850179173,
    227376585863531112677002031251,
    467056170954468301850494793701001,
    36241275390490156321975496980895092369525753,
    284661951906193731091845096405947222295673201 (see examples).
(End)

			5 = (2^4 - 1)/(2^2 - 1)= 11_{2^2} = 11_4.
17 = (2^8 - 1)/(2^4 - 1) = 11_{2^4} = 11_16.
257 = (2^16 - 1)/(2^8 - 1) = 11_{2^8} = 11_256.
757 = (3^9 - 1)/(3^3 - 1) = 111_{3^3} = 111_27.
262657 = (2^27 - 1)/(2^9 - 1) = 111_{2^9} = 111_512.
655357 = (2^32 - 1)/(2^16 - 1) = 11_{2^16} = 11_655356.
4432676798593 = (2^49 - 1)/(2^7 - 1) = 1111111_{2^7} = 1111111_128.
5559917315850179173 = (11^27 - 1)/(11^9 - 1) = 111_{11^3} = 111_1331.
227376585863531112677002031251 = (5^49 - 1)/(5^7 - 1) = 1111111_{5^7}.
467056170954468301850494793701001 = (43^25 - 1)/(43^5 - 1) = 11111_{43^5}.
36241275390490156321975496980895092369525753 = (263^27 - 1)/(263^9 - 1).
284661951906193731091845096405947222295673201 = (167^25 - 1)/(167^5 - 1).

Don Reble, Table of n, a(n) for n = 1..50000
Paul T. Bateman and Rosemarie M. Stemmler, Waring's problem for algebraic number fields and primes of the form (p^r-1)/(p^d-1), Illinois J. Math. 6 (1962), pp. 142-156.
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
Index entries for sequences related to Brazilian numbers.

Equals A003424 \ A023195.

Cf. A085104, A220627.

a(9)-a(16) from Don Reble, Jul 28 2022

a(17)-a(20) from Don Reble, Mar 21 2023

A198244 Primes of the form k^10 + k^9 + k^8 + k^7 + k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 where k is nonprime.

Original entry on oeis.org

11, 10778947368421, 17513875027111, 610851724137931, 614910264406779661, 22390512687494871811, 22793803793211153712637, 79905927161140977116221, 184251916941751188170917, 319465039747605973452001, 1311848376806967295019263, 1918542715220370688851293

Offset: 1

Jonathan Vos Post, Dec 21 2012

nonn

Subsequence of A060885.
From Bernard Schott, Nov 01 2019: (Start)
These are the primes associated with the terms k of A308238.
A162861 = A286301 Union {this sequence}.
The numbers of this sequence R_11 = 11111111111_k with k > 1 are Brazilian primes, so belong to A085104. (End)

			10778947368421 is in the sequence since 10778947368421 = 20^10 + 20^9 + 20^8 + 20^7 + 20^6 + 20^5 + 20^4 + 20^3 + 20^2 + 20 + 1, 20 is not prime, and 10778947368421 is prime.

Cf. A162861, A000040, A088548, A192321, A102909, A060885, A308238.

Similar to A185632 (k^2+ ...), A193366 (k^4+ ...), A194194 (k^6+ ...).

[a: n in [0..500] | not IsPrime(n) and IsPrime(a) where a is (n^10+n^9+n^8+n^7+n^6+n^5+n^4+n^3+n^2+n+1)]; // Vincenzo Librandi, Nov 09 2014

f:= proc(n)
local p,j;
if isprime(n) then return NULL fi;
p:= add(n^j,j=0..10);
if isprime(p) then p else NULL fi
end proc:
map(f, [$1..1000]); # Robert Israel, Nov 19 2014

forcomposite(n=0,10^3,my(t=sum(k=0,10,n^k));if(isprime(t),print1(t,", "))); \\ Joerg Arndt, Nov 10 2014

from sympy import isprime
A198244_list, m = [], [3628800, -15966720, 28828800, -27442800, 14707440, -4379760, 665808, -42240, 682, 0, 1]
for n in range(1,10**4):
    for i in range(10):
        m[i+1]+= m[i]
    if not isprime(n) and isprime(m[-1]):
        A198244_list.append(m[-1]) # Chai Wah Wu, Nov 09 2014

{A060885(A018252(n)) which are in A000040}.

a(5)-a(6) from Robert G. Wilson v, Dec 21 2012
a(7) from Michael B. Porter, Dec 27 2012
Corrected terms a(6)-a(7) and added terms by Chai Wah Wu, Nov 09 2014

cp's OEIS Frontend

A193366 Primes of the form n^4 + n^3 + n^2 + n + 1 where n is nonprime.

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A285642 Smallest Brazilian prime in base n, or 0 if no such prime exists.

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A286094 Nonprime numbers n such that n^4 + n^3 + n^2 + n + 1 is prime.

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A306845 Sophie Germain primes which are Brazilian.

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A306849 Brazilian primes that are also the lesser of a pair of twin primes.

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A306889 Brazilian primes that are also the greater of a pair of twin primes.

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A340796 a(n) is the smallest number with exactly n divisors that are Brazilian.

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A086930 Smallest b>1 such that in base b representation the n-th prime is a repunit.

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