A076481 -id:A076481 - cp's OEIS frontend

A028491 Numbers k such that (3^k - 1)/2 is prime.

3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, 7973131, 8530117

Offset: 1

N. J. A. Sloane, Jean-Yves Perrier (nperrj(AT)ascom.ch)

If k is in the sequence and m=3^(k-1) then m is a term of A033632 (phi(sigma(m)) = sigma(phi(m))), so 3^(A028491-1) is a subsequence of A033632. For example since 9551 is in the sequence, phi(sigma(3^9550)) = sigma(phi(3^9550)). - Farideh Firoozbakht, Feb 09 2005
Salas lists these, except 3, in "Open Problems" p. 6 [March 2012], and proves that the Cantor primes > 3 are exactly the prime-valued cyclotomic polynomials of the form Phi_s(3^{s^j}) == 1 (mod 4).
Also, k such that 3^k-1 is a semiprime - see also A080892. - M. F. Hasler, Mar 19 2013

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 236.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antal Bege and Kinga Fogarasi, Generalized perfect numbers, arXiv:1008.0155 [math.NT], 2010. See p. 81.
Paul Bourdelais, A Generalized Repunit Conjecture, Posting in NMBRTHRY@LISTSERV.NODAK.EDU, Jun 25, 2009.
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Lifchitz, Mersenne and Fermat primes field
Christian Salas, Cantor Primes as Prime-Valued Cyclotomic Polynomials, arXiv:1203.3969 [math.NT], 2012.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
Index to primes in various ranges, form ((k+1)^n-1)/k

Cf. A076481, A033632, A112646.

Do[If[PrimeQ[(3^n-1)/2], Print[n]], {n, 10000}] (* Farideh Firoozbakht, Feb 09 2005 *)

forprime(p=2,1e5,if(ispseudoprime(3^p\2),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

a(13) from Farideh Firoozbakht, Mar 27 2005
a(14)-a(16) from Robert G. Wilson v, Apr 11 2005
All larger terms only correspond to probable primes.
a(17) from Paul Bourdelais, Feb 08 2010
a(18) from Paul Bourdelais, Jul 06 2010
a(19) from Paul Bourdelais, Feb 05 2019
a(20) and a(21) from Ryan Propper, Dec 29 2021
a(22) from Ryan Propper, Nov 06 2023
a(23) from Ryan Propper, Nov 09 2023

A085104 Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1.

Original entry on oeis.org

7, 13, 31, 43, 73, 127, 157, 211, 241, 307, 421, 463, 601, 757, 1093, 1123, 1483, 1723, 2551, 2801, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9901, 10303, 11131, 12211, 12433, 13807, 14281, 17293, 19183, 19531, 20023

Offset: 1

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jul 03 2003

Primes that are base-b repunits with three or more digits for at least one b >= 2: Primes in A053696. Subsequence of A000668 U A076481 U A086122 U A165210 U A102170 U A004022 U ... (for each possible b). - Rick L. Shepherd, Sep 07 2009
From Bernard Schott, Dec 18 2012: (Start)
Also known as Brazilian primes. The primes that are not Brazilian primes are in A220627.
The number of terms k+1 is always an odd prime, but this is not enough to guarantee a prime, for example 111 = 1 + 10 + 100 = 3*37.
The inverses of the Brazilian primes form a convergent series; the sum is slightly larger than 0.33 (see Theorem 4 of Quadrature article in the Links). (End)
It is not known whether there are infinitely many Brazilian primes. See A002383. - Bernard Schott, Jan 11 2013
Primes of the form (n^p - 1)/(n - 1), where p is odd prime and n > 1. - Thomas Ordowski, Apr 25 2013
Number of terms less than 10^n: 1, 5, 14, 34, 83, 205, 542, 1445, 3880, 10831, 30699, 88285, ..., . - Robert G. Wilson v, Mar 31 2014
From Bernard Schott, Apr 08 2017: (Start)
Brazilian primes fall into two classes:
1) when n is prime, we get sequence A023195 except 3 which is not Brazilian,
2) when n is composite, we get sequence A285017. (End)
The conjecture proposed in Quadrature "No Sophie Germain prime is Brazilian (prime)" (see link Bernard Schott, Quadrature, Conjecture 1, page 36) is false. Thanks to Giovanni Resta, who found that a(856) = 28792661 = 1 + 73 + 73^2 + 73^3 + 73^4 = (11111)73 is the 141385th Sophie Germain prime. - _Bernard Schott, Mar 08 2019

			13 is a term since it is prime and 13 = 1 + 3 + 3^2 = 111_3.
31 is a term since it is prime and 31 = 1 + 2 + 2^2 + 2^3 + 2^4 = 11111_2.
From _Hartmut F. W. Hoft_, May 08 2017: (Start)
The sequence represented as a sparse matrix with the k-th column indexed by A006093(k+1), primes minus 1, and row n by A000027(n+1). Traversing the matrix by counterdiagonals produces a non-monotone ordering.
    2    4      6        10             12          16
2  7    31     127      -              8191        131071
3  13   -      1093     -              797161      -
4  -    -      -        -              -           -
5  31   -      19531    12207031       305175781   -
6  43   -      55987    -              -           -
7  -    2801   -        -              16148168401 -
8  73   -      -        -              -           -
9  -    -      -        -              -           -
10  -    -      -        -              -           -
11  -    -      -        -              -           50544702849929377
12  157  22621  -        -              -           -
13  -    30941  5229043  -              -           -
14  211  -      8108731  -              -           -
15  241  -      -        -              -           -
16 -    -      -        -              -           -
17  307  88741  25646167 2141993519227  -           -
18  -    -      -        -              -           -
19  -    -      -        -              -           -
20  421  -      -        10778947368421 -           689852631578947368421
21  463  -      -        17513875027111 -           1502097124754084594737
22  -    245411 -        -              -           -
23  -    292561 -        -              -           -
24  601  346201 -        -              -           -
Except for the initial values in the respective sequences the rows and columns as labeled in the matrix are:
column  2:  A002383            row 2:  A000668
column  4:  A088548            row 3:  A076481
column  6:  A088550            row 4:  -
column 10:  A162861            row 5:  A086122.
(End)

Daniel Lignon, Dictionnaire de (presque) tous les nombres entiers, Ellipses, Paris, 2012, page 174.

Jon E. Schoenfield, Table of n, a(n) for n = 1..10831 (terms up to 10^10; terms 1..3880 from T. D. Noe)
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A189891 (complement), A125134 (Brazilian numbers), A220627 (Primes that are non-Brazilian).
Cf. A003424 (n restricted to prime powers).
Cf. A053696, A086930, A059055.
Equals A023195 \3 Union A285017 with empty intersection.
Primes of the form (b^k-1)/(b-1) for b=2: A000668, b=3: A076481, b=5: A086122, b=6: A165210, b=7: A102170, b=10: A004022.
Primes of the form (b^k-1)/(b-1) for k=3: A002383, k=5: A088548, k=7: A088550, k=11: A162861.

a085104 n = a085104_list !! (n-1)
a085104_list = filter ((> 1) . a088323) a000040_list
-- Reinhard Zumkeller, Jan 22 2014

max = 140; maxdata = (1 - max^3)/(1 - max); a = {}; Do[i = 1; While[i = i + 2; cc = (1 - m^i)/(1 - m); cc <= maxdata, If[PrimeQ[cc], a = Append[a, cc]]], {m, 2, max}]; Union[a] (* Lei Zhou, Feb 08 2012 *)
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]]]; Select[ Range[2, 60], 1 + f@# != Prime@# &] (* Robert G. Wilson v, Mar 31 2014 *)

list(lim)=my(v=List(),t,k);for(n=2,sqrt(lim), t=1+n;k=1; while((t+=n^k++)<=lim,if(isprime(t), listput(v,t))));vecsort(Vec(v),,8) \\ Charles R Greathouse IV, Jan 08 2013

A085104_vec(N,L=List())=forprime(K=3,logint(N+1,2),for(n=2,sqrtnint(N-1,K-1),isprime((n^K-1)\(n-1))&&listput(L,(n^K-1)\(n-1))));Set(L) \\ M. F. Hasler, Jun 26 2018

A010051(a(n)) * A088323(a(n)) > 1. - Reinhard Zumkeller, Jan 22 2014

More terms from David Wasserman, Jan 26 2005

A084738 Smallest prime of the form (n^k-1)/(n-1), or 0 if no such prime exists.

Original entry on oeis.org

3, 13, 5, 31, 7, 2801, 73, 0, 11, 50544702849929377, 13, 30941, 211, 241, 17, 307, 19, 109912203092239643840221, 421, 463, 23, 292561, 601, 0, 321272407, 757, 29, 732541, 31, 917087137, 0, 1123, 2458736461986831391

Offset: 2

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 15 2003

nonn

As mentioned by Dubner, when n is a power (greater than 1) of a prime, then (n^k-1)/(n-1) will usually be composite for all k, which is the case for n = 9, 25, 32 and 49. - T. D. Noe, Jan 23 2004

Here, a(n) is the smallest prime of the form (n^k-1)/(n-1) with k >= 2 while in A285642 it is the smallest prime with k > 2. Differences occur when (n^2-1)/(n-1) = n+1 is prime, and therefore, when n = prime(m) - 1 is in A006093 (see formula). - Bernard Schott, Mar 16 2023

			a(8) = 73 = (8^3-1)/(8-1).

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

Cf. A076481.
Cf. A084740 (least k such that (n^k-1)/(n-1) is prime).
Cf. A006093, A285642.

Table[SelectFirst[(n^# - 1)/(n - 1) & /@ Range[10^3], PrimeQ] /. k_ /; MissingQ@ k -> 0, {n, 2, 34}] (* Michael De Vlieger, Apr 24 2017, Version 10.2 *)

a(A006093(n)) = prime(n) for n >=2. - Bernard Schott, Mar 16 2023

More terms from T. D. Noe, Jan 23 2004

A086122 Primes of the form (5^k-1)/4.

Original entry on oeis.org

31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531, 35032461608120426773093239582247903282006548546912894293926707097244777067146515037165954709053039550781

Offset: 1

Labos Elemer, Jul 23 2003

nonn

Corresponding exponents k are listed in A004061. - Alexander Adamchuk, Jan 23 2007

Cf. A000668, A076481.

Cf. A003463, A004061, A074479.

Do[f=(5^n-1)/4;If[PrimeQ[f],Print[{n,f}]],{n,1,1000}] (* Alexander Adamchuk, Jan 23 2007 *)
Select[(5^Range[300]-1)/4,PrimeQ] (* Harvey P. Dale, Dec 11 2016 *)

a(n) = (5^A004061(n) - 1)/4 = A003463[ A004061(n) ]. - Alexander Adamchuk, Jan 23 2007

A003464 INTERSECT A000040.

More terms from Alexander Adamchuk, Jan 23 2007

A075081 Perfect powers q (A001597) such that (q-1)/2 is prime.

Original entry on oeis.org

27, 2187, 1594323, 7509466514979724803946715958257547, 13915193059764305937984450503671774362956903094027

Offset: 1

Zak Seidov, Oct 16 2002

It can be shown that q must be of the form 3^e with e prime.

The next terms are 3^541, 3^1091, 3^1367, 3^1627, 3^4177, 3^9011, 3^9551, ...

Bo Gyu Jeong, Table of n, a(n) for n = 1..9

Cf. A075079, A076481, A028491.

Extended by Dean Hickerson, Oct 16 2002

A129733 List of primitive prime divisors of the numbers (3^k-1)/2 (A003462) for k>=2, in order of their occurrence.

Original entry on oeis.org

2, 13, 5, 11, 7, 1093, 41, 757, 61, 23, 3851, 73, 797161, 547, 4561, 17, 193, 1871, 34511, 19, 37, 1597, 363889, 1181, 368089, 67, 661, 47, 1001523179, 6481, 8951, 391151, 398581, 109, 433, 8209, 29, 16493, 59, 28537, 20381027, 31, 271, 683

Offset: 1

N. J. A. Sloane, May 13 2007

nonn

Read A003462 term-by-term, factorize each term, write down any primes not seen before.

Except for k=1, there is at least one primitive prime divisor for every k. - T. D. Noe, Mar 01 2010

Max Alekseyev, Primes for k <= 690 (primes for k <= 500 from T. D. Noe)
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., 3 (1892), 265-284.

Cf. A003462, A076481, A028491.

If 3 is replaced with 2, we get A000225, A000043, A108974 respectively.

# produce sequence
s1:=(a,b,M)->[seq( (a^n-b^n)/(a-b),n=0..M)];
# find primes and their indices
s2:=proc(s) local t1,t2,i; t1:=[]; t2:=[];
for i from 1 to nops(s) do if isprime(s[i]) then
t1:=[op(t1),s[i]];
t2:=[op(t2),i-1]; fi; od; RETURN(t1,t2); end;
# get primitive prime divisors in order
s3:=proc(s) local t2,t3,i,j,k,np; t2:=[]; np:=0;
for i from 1 to nops(s) do t3:=ifactors(s[i])[2];
for j from 1 to nops(t3) do p := t3[j][1]; new:=1;
for k from 1 to np do if p = t2[k] then new:= -1; break; fi; od;
if new = 1 then np:=np+1; t2:=[op(t2),p]; fi; od; od;
RETURN(t2); end;

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.

A193574 Smallest divisor of sigma(n) that does not divide n.

Original entry on oeis.org

3, 2, 7, 2, 4, 2, 3, 13, 3, 2, 7, 2, 3, 2, 31, 2, 13, 2, 3, 2, 3, 2, 5, 31, 3, 2, 8, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 3, 2, 4, 2, 3, 2, 3, 2, 31, 3, 3, 2, 7, 2, 4, 2, 3, 2, 3, 2, 7, 2, 3, 2, 127, 2, 4, 2, 3, 2, 3, 2, 5, 2, 3, 2, 5, 2, 4, 2, 3

Offset: 2

Franklin T. Adams-Watters, Aug 27 2011

nonn

a(n) = 2 iff n is an odd number that is not a perfect square.
From Hartmut F. W. Hoft, May 05 2017: (Start)
(1)  Every a(n) > n is a prime: Because of the minimality of a(n), a(n) = u*v with gcd(u,v)=1 leads to the contradiction (u*v)|n. Similarly, a(n)=p^k with p prime an k>1 leads to the contradiction (p^k-1)/(p-1) | n.
(2)  n=p^(2*k), k>=1 and 2*k+1 prime, when a(n) = sigma(n) for n>2: Because n having two distinct prime factors implies sigma(n) composite, and if n is an odd power of a prime then 2|sigma(n). Finally, if 2*k+1=u*v  with  u,v > 1 then sigma(p^(u-1)) divides sigma(p^(2*k)), but not p^(2k), for any prime p, contradicting minimality of a(n). For example, no number sigma(p^8) for any prime p is in the sequence.
(3)  The converse of (2) is false since, e.g. sigma(7^2) = 3*19 so that a(7^2) = 3, and sigma(2^10) = 23*89 so that a(2^10) = 23.
(4)  Conjecture: a(n) > n implies a(n) = sigma(n); tested through n = 20000000.
(5)  Subsequences are: A053183 (sigma(p^2) is prime for prime p), A190527 (sigma(p^4) is prime for prime p), A194257 (sigma(p^6) is prime for prime p), A286301 (sigma(p^10) is prime for prime p)
(6)  Subsequences are: A000668 (primes of form 2^p-1), A076481 (primes of form (3^p-1)/2), A086122 (primes of form (5^p-1)/4), A102170 (primes of form (7^p-1)/6), all when p is prime.
(End)
Up to n = 10^6, there are 89 distinct elements. For those n, a(n) is prime. If it's not, it's a power of 2, a power of 3 or a perfect square <= 121. - David A. Corneth, May 10 2017

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000

Cf. A000203, A007978, A027750, A135718.

Subsequences: A000668, A053183, A076481, A086122, A102170, A190527, A194257, A286301.

import Data.List ((\\))
a193574 n = head [d | d <- [1..sigma] \\ nDivisors, mod sigma d == 0]
   where nDivisors = a027750_row n
         sigma = sum nDivisors
-- Reinhard Zumkeller, May 20 2015, Aug 28 2011

a193574[n_] := First[Select[Divisors[DivisorSigma[1, n]], Mod[n, #]!=0&]]
Map[a193574, Range[2, 80]] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)

a(n)=local(ds);ds=divisors(sigma(n));for(k=2,#ds,if(n%ds[k],return(ds[k])))

cp's OEIS Frontend

A028491 Numbers k such that (3^k - 1)/2 is prime.

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A085104 Primes of the form 1 + n + n^2 + n^3 + ... + n^k, n > 1, k > 1.

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A084738 Smallest prime of the form (n^k-1)/(n-1), or 0 if no such prime exists.

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A086122 Primes of the form (5^k-1)/4.

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A075081 Perfect powers q (A001597) such that (q-1)/2 is prime.

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A129733 List of primitive prime divisors of the numbers (3^k-1)/2 (A003462) for k>=2, in order of their occurrence.

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

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