A240765 -id:A240765 - cp's OEIS frontend

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

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

A242797 Numbers n such that (45^n - 1)/44 is prime.

Original entry on oeis.org

19, 53, 167, 3319, 11257, 34351, 216551

Offset: 1

Robert Price, May 22 2014

a(7) > 10^5.
Numbers corresponding to a(4)-a(6) are probable primes.
All terms are prime.

Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Henri Lifchitz, Mersenne and Fermat primes field

Cf. A028491, A004061, A004062, A004063, A004023, A005808, A004064, A016054, A006032, A006033, A006034, A006035, A127995, A127996, A127997, A127998, A127999, A098438, A128002, A128003, A128004, A128005, A240765.

A242797:=n->`if`(isprime((45^n - 1)/44), n, NULL); seq(A242797(n), n=1..100000); # Wesley Ivan Hurt, Apr 12 2014

Select[Prime[Range[100000]], PrimeQ[(45^# - 1)/44] &]

is(n)=ispseudoprime((45^n-1)/44) \\ Charles R Greathouse IV, Feb 20 2017

a(7)=216551 corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020

A243279 Numbers n such that (46^n - 1)/45 is prime.

Original entry on oeis.org

2, 7, 19, 67, 211, 433, 2437, 2719, 19531

Offset: 1

Robert Price, Jun 02 2014

a(10) > 10^5.
Numbers corresponding to a(7)-a(9) are probable primes.
All terms are prime.

Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Henri Lifchitz, Mersenne and Fermat primes field

A243279:=n->`if`(isprime((46^n - 1)/45), n, NULL); seq(A243279(n), n=1..100000); # Wesley Ivan Hurt, Apr 12 2014

Select[Prime[Range[100000]], PrimeQ[(46^# - 1)/45] &]

is(n)=ispseudoprime((46^n-1)/45) \\ Charles R Greathouse IV, May 22 2017

A245237 Numbers k such that (48^k - 1)/47 is prime.

Original entry on oeis.org

19, 269, 349, 383, 1303, 15031, 200443, 343901

Offset: 1

Robert Price, Jul 14 2014

a(7) > 10^5.

All terms are prime.

Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Henri Lifchitz, Mersenne and Fermat primes field

A245237:=n->`if`(isprime((48^n - 1)/47), n, NULL); seq(A245237(n), n=1..100000); # Wesley Ivan Hurt, Apr 12 2014

Select[Prime[Range[100000]], PrimeQ[(48^# - 1)/47] &]

is(n)=ispseudoprime((48^n-1)/47) \\ Charles R Greathouse IV, Jun 06 2017

a(7) corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020

a(8) from Paul Bourdelais, Mar 03 2025

A245442 Numbers n such that (50^n - 1)/49 is prime.

Original entry on oeis.org

3, 5, 127, 139, 347, 661, 2203, 6521, 210319

Offset: 1

Robert Price, Jul 22 2014

a(9) > 10^5.

All terms are prime.

Harvey Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
Henri Lifchitz, Mersenne and Fermat primes field

A245442:=n->`if`(isprime((50^n - 1)/49), n, NULL); seq(A245442(n), n=1..100000); # Wesley Ivan Hurt, Apr 12 2014

Select[Prime[Range[100000]], PrimeQ[(50^# - 1)/49] &]

is(n)=ispseudoprime((50^n-1)/49) \\ Charles R Greathouse IV, Jun 13 2017

a(9)=210319 corresponds to a probable prime discovered by Paul Bourdelais, Aug 04 2020

A294722 Numbers k such that (44^k - 1)/43 is prime.

Original entry on oeis.org

5, 31, 167, 100511

Offset: 1

Paul Bourdelais, Nov 07 2017

The number corresponding to a(4) is a probable prime.

These are the indices of base-44 repunit primes, i.e., numbers k such that A002275(k) interpreted as a base-44 number and converted to decimal is prime. - Felix Fröhlich, Nov 08 2017

P. Bourdelais, A generalized repunit conjecture

ParallelMap[ If[ PrimeQ[(44^# - 1)/43], #, Nothing] &, Prime@Range @ 10000] (* Robert G. Wilson v, Nov 25 2017 *)

is(n) = ispseudoprime((44^n-1)/43) \\ Felix Fröhlich, Nov 08 2017

ABC2 (44^$a-1)/43 // -f{2*$a}
a: primes from 2 to 1000000

cp's OEIS Frontend

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

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A242797 Numbers n such that (45^n - 1)/44 is prime.

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Maple

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A243279 Numbers n such that (46^n - 1)/45 is prime.

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Maple

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A245237 Numbers k such that (48^k - 1)/47 is prime.

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A245442 Numbers n such that (50^n - 1)/49 is prime.

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Maple

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A294722 Numbers k such that (44^k - 1)/43 is prime.

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