A004061 -id:A004061 - cp's OEIS frontend

A128338 Numbers k such that (8^k + 5^k)/13 is prime.

7, 19, 167, 173, 223, 281, 21647

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.

a(8) > 10^5. - Robert Price, Jan 21 2013

Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128339, A128340, A128341, A128342, A128343. Cf. A004061, A082182, A121877, A059802. Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354.

k=8; Do[p=Prime[n]; f=(k^p+5^p)/(k+5); If[ PrimeQ[f], Print[p] ], {n,1,100}]

is(n)=isprime((8^n+5^n)/13) \\ Charles R Greathouse IV, Feb 17 2017

a(7) from Robert Price, Jan 21 2013

A128343 Numbers k such that (14^k + 5^k)/19 is prime.

Original entry on oeis.org

3, 7, 17, 79, 17477, 19319, 49549

Offset: 1

Alexander Adamchuk, Feb 27 2007

All terms are primes.

a(8) > 10^5. - Robert Price, May 20 2013

Cf. A057171, A082387, A122853, A128335, A128336, A128337, A128338, A128339, A128340, A128341, A128342.
Cf. A004061, A082182, A121877, A059802.
Cf. A062572, A128344, A128345, A128346, A128347, A128348, A128349, A128350, A128351, A128352, A128353, A128354.

k=14; Do[p=Prime[n]; f=(k^p+5^p)/(k+5); If[ PrimeQ[f], Print[p] ], {n,1,100}]

is(n)=isprime((14^n+5^n)/19) \\ Charles R Greathouse IV, Feb 17 2017

a(5)-a(7) from Robert Price, May 20 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

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

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

A096176 Numbers k such that (k^3-1)/(k-1) is prime.

Original entry on oeis.org

2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119, 131, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 167, 168, 173, 176, 188, 189, 192, 194, 203, 206, 209, 215, 218

Offset: 1

Hugo Pfoertner, Jun 22 2004

nonn

Numbers k > 1 such that k^2 + k + 1 is a prime. - Vincenzo Librandi, Nov 16 2010

Therefore essentially the same as A002384. - Georg Fischer, Oct 06 2018

			a(5) = 8 because (8^3-1)/(8-1) = 511/7 = 73 is prime.

Hugo Pfoertner, Table of n, a(n) for n = 1..10000

Cf. A096174 (n^3+1)/(n+1) is prime, A081257 largest prime factor of n^3-1, A096175 n^3-1 is an odd semiprime.
Cf. A028491, A004061. - Daniel McCandless (dkmccandless(AT)gmail.com), Aug 31 2009
Cf. A002384.

[n: n in [2..220] |IsPrime((n^3-1) div (n -1))]; // Vincenzo Librandi, Oct 07 2018

Select[Range[2,550],PrimeQ[(#^3-1)/(#-1)]&] (* Harvey P. Dale, Sep 10 2009 *)

is(n)=isprime(n^2+n+1) \\ Charles R Greathouse IV, Jun 05 2017

3 and 5 added by Daniel McCandless (dkmccandless(AT)gmail.com), Aug 31 2009

Corrected terms, including many previously omitted terms, from Harvey P. Dale, Sep 10 2009

A137410 a(n) = (5^n - 3)/2.

Original entry on oeis.org

-1, 1, 11, 61, 311, 1561, 7811, 39061, 195311, 976561, 4882811, 24414061, 122070311, 610351561, 3051757811, 15258789061, 76293945311, 381469726561, 1907348632811, 9536743164061, 47683715820311, 238418579101561, 1192092895507811, 5960464477539061, 29802322387695311, 149011611938476561

Offset: 0

Ctibor O. Zizka, Apr 15 2008

Sequence is a(n) = a(n;5,3,1) where a(n;A,B,r) = (A^n - B^r)/(A - B) for arbitrary integers A, B, r with A != B.
Primes of this form are sometimes of interest, examples:
A=2, B=1, r=1 gives A000225 and subsequence of primes: A001348,
A=3, B=1, r=1 gives A003462 and subsequence of primes: A028491,
A=3, B=2, r=1 gives A058481 and subsequence of primes: A014224,
A=4, B=1, r=1 gives A002450,
A=4, B=2, r=1 gives A083420,
A=4, B=2, r=2 gives A002446,
A=5, B=1, r=1 gives A003463 and subsequence of primes: A004061,
A=5, B=2, r=1 gives A037577.
Sum of n-th row of triangle of powers of 5: 1; 5 1 5; 25 5 1 5 25; 125 25 5 1 5 25 125; ... (cf. Examples). - Philippe Deléham, Feb 24 2014
Integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n (see Campbell and Zujev). - Michel Marcus, Mar 02 2016

			From _Philippe Deléham_, Feb 24 2014: (Start)
a(1) = 1;
a(2) = 5 + 1 + 5 = 11;
a(3) = 25 + 5 + 1 + 5 + 25 = 61;
a(4) = 125 + 25 + 5 + 1 + 5 + 25 + 125 = 311;
etc. (End)

M. F. Hasler, Table of n, a(n) for n = 0..100
Geoffrey B Campbell and Aleksander Zujev, On integer solutions to x^5 - (x+1)^5 -(x+2)^5 +(x+3)^5 = 5^m + 5^n, arXiv:1603.00080 [math.NT], 2016.
Index entries for linear recurrences with constant coefficients, signature (6,-5).

Cf. A000040, A000225, A001348, A002446, A002450, A003462, A003463, A004061, A014224, A024049, A028491, A037577, A058481, A083420, A125831.

[(5^n-3)/2: n in [0..25]]; // Vincenzo Librandi, Mar 02 2016

LinearRecurrence[{6, -5}, {1, 11}, 25] (* Vincenzo Librandi, Mar 02 2016 *)
(5^Range[30]-3)/2 (* Harvey P. Dale, Feb 24 2017 *)

a(n) = (5^n - 3) / 2; \\ Michel Marcus, Mar 02 2016

a(n) = (5^n - 3)/2.
From Colin Barker, May 01 2012: (Start)
a(n) = 6*a(n-1) - 5*a(n-2).
G.f.: (-1+7*x)/((1-x)*(1-5*x)). (End)
a(n) = 5*a(n-1) + 6, a(1) = 1. - Philippe Deléham, Feb 24 2014
From Elmo R. Oliveira, Dec 11 2023: (Start)
a(n) = A024049(n)/2 - 1 = A125831(n) - 1.
E.g.f.: (1/2)*(exp(5*x) - 3*exp(x)). (End)

More terms from Michel Marcus, Mar 02 2016

Edited and missing term a(0) inserted by M. F. Hasler, Jul 10 2018

cp's OEIS Frontend

A128338 Numbers k such that (8^k + 5^k)/13 is prime.

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A128343 Numbers k such that (14^k + 5^k)/19 is prime.

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

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

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

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