A005808 -id:A005808 - cp's OEIS frontend

A028489 Duplicate of A005808.

17, 19, 79, 139, 907, 1907, 2029, 4801, 5153, 10867

Offset: 0

Jean-Yves Perrier (nperrj(AT)ascom.ch)

dead

The original definition, "Exponents of generalized rep-units in base 11", lacks the requirement of primality. - M. F. Hasler, Mar 19 2013

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

A240765 Numbers n such that (43^n - 1)/42 is prime.

Original entry on oeis.org

5, 13, 6277, 26777, 27299, 40031, 44773, 194119

Offset: 1

Robert Price, Apr 12 2014

a(8) > 10^5. - Robert Price, Apr 12 2014

H. Dubner, Generalized repunit primes, Math. Comp., 61 (1993), 927-930.
H. 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:=n->`if`(isprime((43^n - 1)/42), n, NULL); seq(A240765(n), n=1..100000); # Wesley Ivan Hurt, Apr 12 2014

Select[Prime[Range[100000]], PrimeQ[(43^#-1)/42]&]

is(n)=ispseudoprime((43^n-1)/42) \\ Charles R Greathouse IV, Feb 20 2017

a(8) from Paul Bourdelais, Aug 04 2020

A117545 Least k such that Phi(k,n), the k-th cyclotomic polynomial evaluated at n, is prime.

Original entry on oeis.org

2, 2, 1, 1, 3, 1, 5, 1, 6, 2, 9, 1, 5, 1, 3, 2, 3, 1, 19, 1, 3, 2, 5, 1, 6, 4, 3, 2, 5, 1, 7, 1, 3, 6, 21, 2, 10, 1, 6, 2, 3, 1, 5, 1, 19, 2, 10, 1, 14, 3, 6, 2, 11, 1, 6, 4, 3, 2, 3, 1, 7, 1, 5, 204, 12, 2, 6, 1, 3, 2, 3, 1, 5, 1, 3, 6, 3, 2, 5, 1, 6, 2, 5, 1, 5, 11, 7, 2, 3, 1, 6, 12, 7, 4, 7, 2, 17, 1, 3

Offset: 1

T. D. Noe, Mar 28 2006

nonn

Note that a(n)=1 iff n-1 is prime because Phi(1,x)=x-1. For n<2048, we have the bound a(n)<251. However, a(2048) is greater than 10000. Is a(n) defined for all n? For fixed n, there are many sequences listing the k that make Phi(k,n) prime: A000043, A028491, A004061, A004062, A004063, A004023, A005808, A016054, A006032, A006033, A006034, A006035.

T. D. Noe, Table of n, a(n) for n = 1..2047

Cf. A117544 (least k such that Phi(n, k) is prime).

Table[k=1; While[ !PrimeQ[Cyclotomic[k,n]], k++ ]; k, {n,100}]

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

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.

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

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, A243279.

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

A273598 Numbers k such that (11^k - 6^k)/5 is prime.

Original entry on oeis.org

2, 3, 11, 163, 191, 269, 1381, 1493, 38453

Offset: 1

Tim Johannes Ohrtmann, May 26 2016

All terms are prime.

The corresponding primes: 17, 223, 56989774711, ...

Cf. A005808, A210506, A128027, A216181, A128347, A273599, A273600, A273601, A062577.

Select[Range[1, 10000], PrimeQ[(11^# - 6^#)/5] &]

for(n=1, 10000, if(isprime((11^n - 6^n)/5), print1(n, ", ")))

a(9) from Michael S. Branicky, Nov 10 2024

A273599 Numbers k such that (11^k - 7^k)/4 is prime.

Original entry on oeis.org

5, 19, 67, 107, 593, 757, 1801, 2243, 2383, 6043, 10181, 11383, 15629

Offset: 1

Tim Johannes Ohrtmann, May 26 2016

All terms are prime.
The corresponding primes: 36061, 15286922888307293287, 1483371444025889427763765389467527889556636442659800720575790059738807, ...
a(14) > 50000. - Michael S. Branicky, Nov 11 2024

Cf. A005808, A210506, A128027, A216181, A128347, A273598, A273600, A273601, A062577.

Select[Range[1, 10000], PrimeQ[(11^# - 7^#)/4] &]

for(n=1, 10000, if(isprime((11^n - 7^n)/4), print1(n, ", ")))

A273600 Numbers k such that (11^k - 8^k)/3 is prime.

Original entry on oeis.org

2, 7, 11, 17, 37, 521, 877, 2423

Offset: 1

Tim Johannes Ohrtmann, May 26 2016

All terms are prime.
The corresponding primes: 19, 5796673, 92240578673, 167731742895202841, 113345629904382710526197539019199125641, ...
a(9) > 50000. - Michael S. Branicky, Nov 11 2024

Cf. A005808, A210506, A128027, A216181, A128347, A273598, A273599, A273601, A062577.

Select[Range[1, 10000], PrimeQ[(11^# - 8^#)/3] &]

for(n=1, 10000, if(isprime((11^n - 8^n)/3), print1(n, ", ")))

cp's OEIS Frontend

A028489 Duplicate of A005808.

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

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A240765 Numbers n such that (43^n - 1)/42 is prime.

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Maple

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A117545 Least k such that Phi(k,n), the k-th cyclotomic polynomial evaluated at n, is prime.

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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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Maple

Mathematica

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A273598 Numbers k such that (11^k - 6^k)/5 is prime.

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