A084740 -id:A084740 - cp's OEIS frontend

A360738 a(n) = A084740(n) - 1.

1, 1, 2, 1, 2, 1, 4, 2, -1, 1, 16, 1, 4, 2, 2, 1, 2, 1, 18, 2, 2, 1, 4, 2, -1, 6, 2, 1, 4, 1, 6, -1, 2, 12, 312, 1, 12, 2, 348, 1, 2, 1, 4, 4, 18, 1, 126, 18, -1, 2, 4228, 1, 10, 2, 16, 6, 2, 1, 2, 1, 6, 2, 4, -1, 18, 1, 18, 4, 2, 1, 2, 1, 4, 4, 2, 40, 2, 1, 4, 2, -1, 1, 4, 16, 4, 10, 6, 1, 2, 2, 4420

Offset: 1

Ya-Ping Lu, Feb 18 2023

sign

a(n) is the smallest k such that 1 + n + n^2 + ... + n^k is a prime. a(n) = -1 if no such k exists.
From Jon E. Schoenfield, Feb 22 2023: (Start)
Let s(n,k) = Sum_{j=0..k} n^j. Then if k+1 is not a prime, s(n,k) is divisible by s(n,j) for some positive j < k, so s(n,k) is not a prime.
Additionally, if n = 2^d (for d >= 1), then if k+1 is a prime, s(n,k) is usually divisible by 2^(k+1)-1. For d <= 16, it seems that the only exceptions are s(4,1) = 5, s(8,2) = 73, s(16,1) = 17, s(32,4) = 601*1801, s(64,1) = 5*13, s(64,2) = 3*19*73, s(128,6) = 4432676798593, s(256,1) = 257, s(512,2) = 262657, s(1024,1) = 5^2*41, s(1024,4) = 251*601*1801*4051, s(2048,10) = 727*p31, s(4096,1) = 17*241, s(4096,2) = 3*19*37*73*109, s(8192,12) = 4057*6740339310641*p31, s(16384,1) = 5*29*113, s(16384,6) = 4363953127297*4432676798593, s(32768,2) = 73*631*23311, s(32768,4) = 601*1801*100801*10567201, s(65536,1) = 65537, and s(131072,16) = 12761663*179058312604392742511009*p52.
So a(32) = -1 because the sum s(32,k) is not prime for any k:
for each k such that k+1 is not a prime, s(32,k) is divisible by s(32,j) for some positive j < k;
for each k such that k+1 is a prime except for k=4, s(32,k) is divisible by 2^(k+1)-1; and
s(32,4) = 601*1801 is a nonprime.
Similarly, a(64) = -1 because the sum s(64,k) is not prime for any k:
for each k such that k+1 is not a prime, s(64,k) is divisible by s(64,j) for some positive j < k;
for each k such that k+1 is a prime except for k=1 and k=2, s(64,k) is divisible by 2^(k+1)-1; and
s(64,1) = 5*13 and s(64,2) = 3*19*73 are nonprime. (End)

Cf. A084740.

a(n, lim=oo)=if((issquare(n)&&n%2==1&&n>1)||n==32||n==64, -1, my(s=1); for(k=1, lim, s+=n^k; if(ispseudoprime(s), return(k)) ); oo) \\ Andrew Howroyd, Feb 22 2023

a((2m+1)^2) = -1 for m >= 1.

a(32)-a(79) from Jon E. Schoenfield, Feb 21 2023

a(80)-a(91) from Andrew Howroyd, Feb 22 2023

A000978 Wagstaff numbers: numbers k such that (2^k + 1)/3 is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 17, 19, 23, 31, 43, 61, 79, 101, 127, 167, 191, 199, 313, 347, 701, 1709, 2617, 3539, 5807, 10501, 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321, 986191, 4031399

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

It is easy to see that the definition implies that k must be an odd prime. - N. J. A. Sloane, Oct 06 2006
The terms from a(32) on only give probable primes as of 2018. Caldwell lists the largest certified primes. - Jens Kruse Andersen, Jan 10 2018
Prime numbers of the form 1+Sum_{i=1..m} 2^(2i-1). - Artur Jasinski, Feb 09 2007
There is a new conjecture stating that a Wagstaff number is prime under the following condition (based on DiGraph cycles under the LLT): Let p be a prime integer > 3, N(p) = 2^p+1 and W(p) = N(p)/3, S(0) = 3/2 (or 1/4) and S(i+1) = S(i)^2 - 2 (mod N(p)). Then W(p) is prime iff S(p-1) == S(0) (mod W(p)). - Tony Reix, Sep 03 2007
As a member of the DUR team (Diepeveen, Underwood, Reix), and thanks to the LLR tool built by Jean Penne, I've found a new and big Wagstaff PRP: (2^4031399+1)/3 is Vrba-Reix PRP! This Wagstaff number has 1,213,572 digits and today is the 3rd biggest PRP ever found. I've done a second verification on a Nehalem core with the PFGW tool. - Tony Reix, Feb 20 2010
13347311 and 13372531 were found to be terms of this sequence (maybe not the next ones) by Ryan Propper in September 2013. - Max Alekseyev, Oct 07 2013
The next term is larger than 10 million. - Gord Palameta, Mar 22 2019
Ryan Propper found another likely term, 15135397, though it only corresponds to a probable prime. - Charles R Greathouse IV, Jul 01 2021

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
S. S. Wagstaff, Jr., personal communication.

J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
C. Caldwell's The Top Twenty, Wagstaff.
C. Caldwell, New Mersenne Conjecture
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930. [Annotated scanned copy]
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
Editor's Note, Table of Wagstaff primes sent by D. H. Lehmer (annotated and scanned copy)
J. E. Foster, Problem 174, A solution in primes, Math. Mag., 27 (1954), 156-157.
R. K. Guy, Letter to N. J. A. Sloane, Aug 1986
R. K. Guy, Letter to N. J. A. Sloane, 1987
H. Lifchitz, Mersenne and Fermat primes field
H. & R. Lifchitz, PRP Top Records.
Henri & Renaud Lifchitz, PRP Records.
Gord Palameta, There are no new Wagstaff primes with exponent below 10 million, mersenneforum.org
Ryan Propper et al., New Wagstaff PRP exponents, mersenneforum.org
PRP top list: PRP top [From _Tony Reix_, Feb 20 2010]
Tony Reix, Yahoo Primeform Group Message 10184 dd. Feb 20, 2010, reconstruction in html.
T. Reix, Some Maths about the Vrba-Reix PRP test [From _Tony Reix_, Feb 20 2010]
Djurre G. Sikkema, Probable primality testing for Wagstaff prime, Bachelor's project mathematics, Univ. Groningen (Netherlands 2024). See p. 32.
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
Eric Weisstein's World of Mathematics, Wagstaff Prime
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Wikipedia, Wagstaff prime
R. G. Wilson, v, Letter to N. J. A. Sloane, circa 1991.

Cf. A107036 (indices of prime Jacobsthal numbers).

Cf. A000979, A124400, A124401, A127955, A127956, A127957, A127958, A127936, A010051, A065091, A001045.

a000978 n = a000978_list !! (n-1)
a000978_list = filter ((== 1) . a010051 . a001045) a065091_list
-- Reinhard Zumkeller, Mar 24 2013

Select[Range[5000], PrimeQ[(2^# + 1)/3] &] (* Michael De Vlieger, Jan 10 2018 *)
Select[Prime[Range[2,500]],PrimeQ[(2^#+1)/3]&] (* Harvey P. Dale, Jun 13 2022 *)

forprime(p=2,5000,if(ispseudoprime(2^p\/3),print1(p", "))) \\ Charles R Greathouse IV, Jul 15 2011

from gmpy2 import divexact
from sympy import prime, isprime
A000978 = [p for p in (prime(n) for n in range(2,10**2)) if isprime(divexact(2**p+1,3))] # Chai Wah Wu, Sep 04 2014

a(n) = A107036(n) for n>1. - Alexander Adamchuk, Feb 10 2007

a(30) from Kamil Duszenko (kdusz(AT)wp.pl), Feb 03 2003; a(30) was proved prime by Francois Morain with FastECPP. - Tony Reix, Sep 03 2007
a(31)-a(39) from Robert G. Wilson v, Apr 11 2005
a(40) from Vincent Diepeveen (diep(AT)xs4all.nl) added by Alexander Adamchuk, Jun 19 2008
a(41) from Tony Reix, Feb 20 2010

A057171 Numbers n such that (5^n+1)/6 is a prime.

Original entry on oeis.org

5, 67, 101, 103, 229, 347, 4013, 23297, 30133, 177337, 193939, 266863, 277183, 335429, 1856147

Offset: 1

N. J. A. Sloane, Sep 15 2000

With the discovery of a(15), the best fit line slope G=0.55167 (see link to Generalized Repunit Conjecture). This sequence is converging nicely to the conjectured slope G=0.56145948. - Paul Bourdelais, Feb 26 2019

Paul Bourdelais, A Generalized Repunit Conjecture
J. Brillhart et al., Factorizations of b^n +- 1, Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 3rd edition, 2002.
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
Eric Weisstein's World of Mathematics, Repunit
R. G. Wilson, v, Letter to N. J. A. Sloane, circa 1991.

a={};Do[x=(5^n+1)/6;If[PrimeQ[x],AppendTo[a,n]],{n,0,12^2}];a (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Range[40000],PrimeQ[(5^#+1)/6]&] (* The program generates the first 9 terms of the sequence. *) (* Harvey P. Dale, Dec 28 2024 *)

isok(n) = (denominator(p=(5^n+1)/6) == 1) && isprime(p); \\ Michel Marcus, Oct 28 2017

More terms from Kamil Duszenko (kdusz(AT)wp.pl), Jun 23 2003
30133 from Herman Jamke (hermanjamke(AT)fastmail.fm), Jan 05 2008
a(10) discovered 10/29/08 is a probable prime based on trial factoring to 3.5e13 and Fermat testing base 2. - Paul Bourdelais, Nov 04 2008
a(11)=193939 from Paul Bourdelais discovered 12/24/08 is a probable prime based on trial factoring to 4e13 and Fermat primality testing base 2. - Paul Bourdelais, Dec 24 2008
a(12)=266863 is a probable prime discovered by Paul Bourdelais, Jul 09 2010
a(13)=277183 is a probable prime discovered by Paul Bourdelais, Jul 16 2010
a(14)=335429 is a probable prime discovered by Paul Bourdelais, Aug 23 2010
a(15)=1856147 corresponds to a probable prime discovered by Paul Bourdelais, Feb 26 2019

A007658 Numbers k such that (3^k + 1)/4 is prime.

Original entry on oeis.org

3, 5, 7, 13, 23, 43, 281, 359, 487, 577, 1579, 1663, 1741, 3191, 9209, 11257, 12743, 13093, 17027, 26633, 104243, 134227, 152287, 700897, 1205459, 1896463, 2533963, 2674381, 7034611

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Prime repunits in base -3.

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Paul Bourdelais, A Generalized Repunit Conjecture
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. [Annotated scanned copy]
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
Robert G. Wilson v, Letter to N. J. A. Sloane, circa 1991.

lst={};Do[If[PrimeQ[(3^n+1)/4], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 21 2008 *)

is(n)=ispseudoprime((3^n+1)/4) \\ Charles R Greathouse IV, Apr 29 2015

a(20) from Robert G. Wilson v, Apr 11 2005
a(22) from Paul Bourdelais, Nov 08 2007
a(23) from Paul Bourdelais, Apr 07 2008
a(24) from Paul Bourdelais, Apr 05 2010
a(25) from Paul Bourdelais, Aug 28 2015
a(26) from Paul Bourdelais, Jan 30 2020
a(27) from Paul Bourdelais, Mar 06 2020
a(28) from Paul Bourdelais, Mar 22 2024
a(29) from Paul Bourdelais, Dec 04 2024

A001562 Numbers n such that (10^n + 1)/11 is a prime.

Original entry on oeis.org

5, 7, 19, 31, 53, 67, 293, 641, 2137, 3011, 268207, 1600787

Offset: 1

N. J. A. Sloane

The a(10) to a(11) gap represents the largest relative gap seen so far in searching repunits with bases between -12 and 12. On average, there should have been 4 more primes added to this sequence by a(11), instead of just 1. - Paul Bourdelais, Feb 11 2010

J. Brillhart et al., Factorizations of b^n +- 1. Contemporary Mathematics, Vol. 22, Amer. Math. Soc., Providence, RI, 2nd edition, 1985; and later supplements.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

P. Bourdelais, A Generalized Repunit Conjecture, 2009.
J. Brillhart, Letter to N. J. A. Sloane, Aug 08 1970
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. [Annotated scanned copy]
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
H. Lifchitz, Mersenne and Fermat primes field
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit
R. G. Wilson, v, Letter to N. J. A. Sloane, circa 1991.

Equals 2*A054416 + 1.

Odd terms of A309358.

Select[Range[3000], PrimeQ[(10^# + 1) / 11] &] (* Vincenzo Librandi, Oct 29 2017 *)

isok(n) = (denominator(p=(10^n+1)/11)==1) && isprime(p); \\ Michel Marcus, Oct 29 2017

a(11) corresponds to a probable prime discovered by Paul Bourdelais, Feb 11 2010

a(12) corresponds to a probable prime discovered by Paul Bourdelais, May 04 2020

A084742 Least k such that (n^k+1)/(n+1) is prime, or 0 if no such prime exists.

Original entry on oeis.org

3, 3, 3, 5, 3, 3, 0, 3, 5, 5, 5, 3, 7, 3, 3, 7, 3, 17, 5, 3, 3, 11, 7, 3, 11, 0, 3, 7, 139, 109, 0, 5, 3, 11, 31, 5, 5, 3, 53, 17, 3, 5, 7, 103, 7, 5, 5, 7, 1153, 3, 7, 21943, 7, 3, 37, 53, 3, 17, 3, 7, 11, 3, 0, 19, 7, 3, 757, 11, 3, 5, 3, 7, 13, 5, 3, 37, 3, 3, 5, 3, 293, 19, 7, 167, 7, 7, 709, 13, 3, 3, 37, 89, 71, 43, 37

Offset: 2

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

nonn

When (n^k+1)/(n+1) is prime, k must be prime. As mentioned by Dubner and Granlund, when n is a perfect power (the power is greater than 2), then (n^k+1)/(n+1) will usually be composite for all k, which is the case for n = 8, 27, 32 and 64. a(n) are only probable primes for n = {53, 124, 150, 182, 205, 222, 296}.
a(n) = 0 if n = {8, 27, 32, 64, 125, 243, ...}. - Eric Chen, Nov 18 2014
More terms: a(124) = 16427, a(150) = 6883, a(182) = 1487, a(205) = 5449, a(222) = 1657, a(296) = 1303. For n up to 300, a(n) is currently unknown only for n = {97, 103, 113, 175, 186, 187, 188, 220, 284}. All other terms up to a(300) are less than 1000. - Eric Chen, Nov 18 2014
a(97) > 31000. - Eric Chen, Nov 18 2014
a(311) = 2707, a(313) = 4451. - Eric Chen, Nov 20 2014
a(n)=3 if and only if n^2-n+1 is a prime; that is, n belongs to A055494. - Thomas Ordowski, Sep 19 2015
From Altug Alkan, Sep 29 2015: (Start)
a(n)=5 if and only if Phi(10, n) is prime and Phi(6, n) is composite. n belongs to A246392.
a(n)=7 if and only if Phi(14, n) is prime, and Phi(10, n) and Phi(6, n) are both composite. n belongs to A250174.
a(n)=11 if and only if Phi(22, n) is prime, and Phi(14, n), Phi(10, n) and Phi(6, n) are all composite. n belongs to A250178.
Where Phi(k, n) is the k-th cyclotomic polynomial. (End)
a(97) > 800000 (or a(97) = 0). - Wang Runsen, May 10 2023

			a(5) = 5 as (5^5 + 1)/(5 + 1) = 1 - 5 + 5^2 - 5^3 + 5^4 = 521 is a prime.
a(7) = 3 as (7^3 + 1)/(7 + 1) = 1 - 7 + 7^2 = 43 is a prime.

Eric Chen, Table of known a(n) up to a(300)
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
Eric Weisstein's World of Mathematics, Repunit
Robert G. Wilson v, Letter to N. J. A. Sloane, circa 1991.

Cf. A084741, A065507, A084740, A128164, A126659, A103795.

a(n) = {l=List([8, 27, 32, 64, 125, 243, 324, 343]); for(q=1, #l, if(n==l[q], return(0))); k=2; while(k, s=(n^prime(k)+1)/(n+1); if(ispseudoprime(s), return(prime(k))); k++)}
n=2; while(n<361, print1(a(n), ", "); n++) \\ Eric Chen, Nov 25 2014

More terms from T. D. Noe, Jan 22 2004

A084741 Smallest prime of the form (n^k+1)/(n+1), or 0 if no such prime exists.

Original entry on oeis.org

3, 7, 13, 521, 31, 43, 0, 73, 9091, 13421, 19141, 157, 7027567, 211, 241, 22796593, 307, 274019342889240109297, 152381, 421, 463, 39700406579747, 183458857, 601, 135938684703251, 0, 757, 574995877

Offset: 2

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

nonn

When n is a power (greater than 2) of a prime, then (n^k+1)/(n+1) will usually be composite for all k, which is the case for n = 8, 27, 32, 64, 125. The next term, a(30), is a 204-digit number. - T. D. Noe, Jan 22 2004

			a(7)= 43 = (7^3 +1 )/(7+1) = 1 - 7 + 7^2 is a prime.

H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
R. G. Wilson, v, Letter to N. J. A. Sloane, circa 1991.

Cf. A084742 (least k such that (n^k+1)/(n+1) is prime).

More terms from T. D. Noe, Jan 22 2004

A128164 Least k > 2 such that (n^k - 1)/(n-1) is prime, or 0 if no such prime exists.

Original entry on oeis.org

3, 3, 0, 3, 3, 5, 3, 0, 19, 17, 3, 5, 3, 3, 0, 3, 25667, 19, 3, 3, 5, 5, 3, 0, 7, 3, 5, 5, 5, 7, 0, 3, 13, 313, 0, 13, 3, 349, 5, 3, 1319, 5, 5, 19, 7, 127, 19, 0, 3, 4229, 103, 11, 3, 17, 7, 3, 41, 3, 7, 7, 3, 5, 0, 19, 3, 19, 5, 3, 29, 3, 7, 5, 5, 3, 41, 3, 3, 5, 3, 0, 23, 5, 17, 5, 11, 7, 61, 3, 3

Offset: 2

Alexander Adamchuk, Feb 20 2007

nonn

a(n) = A084740(n) for all n except n = p-1, where p is an odd prime, for which A084740(n) = 2.
All nonzero terms are odd primes.
a(n) = 0 for n = {4,9,16,25,32,36,49,64,81,100,121,125,144,...}, which are the perfect powers with exceptions of the form n^(p^m) where p>2 and (n^(p^(m+1))-1)/(n^(p^m)-1) are prime and m>=1 (in which case a(n^(p^m))=p). - Max Alekseyev, Jan 24 2009
a(n) = 3 for n in A002384, i.e., for n such that n^2 + n + 1 is prime.
a(152) > 20000. - Eric Chen, Jun 01 2015
a(n) is the least number k such that (n^k - 1)/(n-1) is a Brazilian prime, or 0 if no such Brazilian prime exists. - Bernard Schott, Apr 23 2017
These corresponding Brazilian primes are in A285642. - Bernard Schott, Aug 10 2017
a(152) = 270217, see the top PRP link. - Eric Chen, Jun 04 2018
a(184) = 16703, a(200) = 17807, a(210) = 19819, a(306) = 26407, a(311) = 36497, a(326) = 26713, a(331) = 25033; a(185) > 66337, a(269) > 63659, a(281) > 63421, and there are 48 unknown a(n) for n <= 1024. - Eric Chen, Jun 04 2018
Six more terms found: a(522)=20183, a(570)=12907, a(684)=22573, a(731)=15427, a(820)=12043, a(996)=14629. - Michael Stocker, Apr 09 2020

			a(7) = 5 because (7^5 - 1)/6 = 2801 = 11111_7 is prime and (7^k - 1)/6 = 1, 8, 57, 400 for k = 1, 2, 3, 4. - _Bernard Schott_, Apr 23 2017

Max Alekseyev and Eric Chen, Table of n, a(n) for n = 2..184 (terms 2..151 from Max Alekseyev)
Eric Chen, Table of n, a(n) for n = 2..1024 status (updated by Jinyuan Wang)
H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Richard Fischer, Generalized repunit primes of the form (B^N-1)/(B-1)
Top PRPs, Search by (152^n-1)/(152-1)
Top PRPs, Search by (b^n-1)/a
Eric Weisstein's World of Mathematics, Repunit

Cf. A084738, A065854, A084740, A084741, A065507, A084742, A066180, A084732, A285642, A085104.
Cf. A002384, A049409, A100330, A162862, A217070-A217089. (numbers b such that (b^p-1)/(b-1) is prime for prime p = 3 to 97)
Cf. A000043, A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A133857, A006035, A127995, A127996, A127997, A204940, A127998, A127999, A128000, A181979, A098438, A128002, A209120, A185073, A128003, A128004, A181987, A128005, A239637, A240765, A294722, A242797, A243279, A267375, A245237, A245442, A173767. (numbers n such that (b^n-1)/(b-1) is prime for b = 2 to 53)
A126589 gives locations of zeros.

Table[Function[m, If[m > 0, k = 3; While[! PrimeQ[(m^k - 1)/(m - 1)], k++]; k, 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 *)

a052409(n) = my(k=ispower(n)); if(k, k, n>1)
a052410(n) = if (ispower(n, , &r), r, n)
is(n) = issquare(n) || (ispower(n) && !ispseudoprime((n^a052410(a052409(n))-1)/(n-1)))
a(n) = if(is(n), 0, forprime(p=3, 2^16, if(ispseudoprime((n^p-1)/(n-1)), return(p)))) \\ Eric Chen, Jun 01 2015, corrected by Eric Chen, Jun 04 2018, after Charles R Greathouse IV in A052409 and Michel Marcus in A052410

a(18) = 25667 found by Henri Lifchitz, Sep 26 2007

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

A065854 Smallest prime q such that (p^q-1)/(p-1) is a prime, where p = prime(n).

Original entry on oeis.org

2, 3, 3, 5, 17, 5, 3, 19, 5, 5, 7, 13, 3, 5, 127, 11, 3, 7, 19, 3, 5, 5, 5, 3, 17, 3, 19, 17, 17, 23, 5, 3, 11, 163, 7, 13, 17, 7, 3, 3, 19, 17, 17, 5, 31, 577, 41, 239, 5, 11, 113, 5, 17, 7, 23, 5

Offset: 1

Vladeta Jovovic, Nov 26 2001

a(n) = 2*A065813(n) + 1, n > 1.

Andy Steward, Titanic Prime Generalized Repunits

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

Do[p = Prime[n]; k = 1; While[ !PrimeQ[ (p^Prime[k] - 1)/(p - 1)], k++ ]; Print[ Prime[k]], {n, 1, 56} ]
Table[Module[{q=2},While[!PrimeQ[(p^q-1)/(p-1)],q=NextPrime[q]];q],{p,Prime[Range[60]]}] (* Harvey P. Dale, Aug 05 2025 *)

{ allocatemem(932245000); for (n=1, 100, p=prime(n); q=2; while (!isprime((p^q - 1)/(p - 1)), q=nextprime(q + 1)); write("b065854.txt", n, " ", q) ) } \\ Harry J. Smith, Nov 01 2009

cp's OEIS Frontend

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A084742 Least k such that (n^k+1)/(n+1) is prime, or 0 if no such prime exists.

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A084741 Smallest prime of the form (n^k+1)/(n+1), or 0 if no such prime exists.

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A128164 Least k > 2 such that (n^k - 1)/(n-1) is prime, or 0 if no such prime exists.