A002384 -id:A002384 - 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

A185632 Primes of the form n^2 + n + 1 where n is nonprime.

Original entry on oeis.org

3, 43, 73, 157, 211, 241, 421, 463, 601, 757, 1123, 1483, 2551, 2971, 3307, 3907, 4423, 4831, 5701, 6007, 6163, 6481, 8191, 9901, 11131, 12211, 12433, 13807, 14281, 19183, 20023, 20593, 21757, 22651, 23563, 24181, 26083, 26407, 27061, 28393, 31153, 35533

Offset: 1

Bernard Schott, Dec 18 2012

nonn

These are the primes associated with A182253.
All these numbers are in A002383 but not in A053183.
All the numbers n^2 + n + 1 = 111_n with n >= 2 are by definition Brazilian numbers: A125134. See Links: "Les nombres brésiliens" - Section V.5 page 35.

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38.
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, avril-juin 2010, pages 30-38, included here with permission from the editors of Quadrature.

Cf. A002383, A002384, A053182, A053183, A182253.

Cf. A085104, A125134.

Select[Table[If[PrimeQ[n],Nothing,n^2+n+1],{n,200}],PrimeQ] (* Harvey P. Dale, Apr 02 2023 *)

lista(nn) = {for (n = 1, nn, if (! isprime(n) && isprime(p = n^2 + n + 1), print1(p, ", ");););} \\ Michel Marcus, Sep 04 2013

A217071 Numbers k such that (k^17-1)/(k-1) is prime.

Original entry on oeis.org

2, 11, 20, 21, 28, 31, 55, 57, 62, 84, 87, 97, 107, 109, 129, 147, 149, 157, 160, 170, 181, 189, 191, 207, 241, 247, 251, 274, 295, 297, 315, 327, 335, 349, 351, 355, 364, 365, 368, 379, 383, 410, 419, 423, 431, 436, 438, 466, 472, 506, 513, 527, 557, 571, 597

Offset: 1

Tim Johannes Ohrtmann, Sep 26 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002384, A049409, A100330, A162862, A217070-A217089.

[n: n in [2..1000] |IsPrime((n^17 - 1) div (n - 1))]; // Vincenzo Librandi, Sep 28 2012

Select[Range[2, 1000], PrimeQ[(#^17 - 1)/(# - 1)] &] (* T. D. Noe, Sep 26 2012 *)

is(n)=isprime((n^17-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

More terms from T. D. Noe, Sep 26 2012

A110284 Squares of the form 4p - 3, where p is a prime.

Original entry on oeis.org

9, 25, 49, 121, 169, 289, 625, 841, 961, 1225, 1681, 1849, 2401, 3025, 4489, 5929, 6889, 10201, 11881, 13225, 14161, 15625, 17689, 19321, 20449, 22801, 24025, 24649, 25921, 32041, 32761, 39601, 41209, 44521, 48841, 49729, 55225, 57121, 69169

Offset: 1

Giovanni Teofilatto, Sep 07 2005

nonn

Also: squares of the form 2*s-3, where s is a semiprime, A107317. - Franklin T. Adams-Watters, Jun 28 2010

Squares are less dense then primes and easy to generate so it's faster to check squares if they are of the required form than to check if primes are of the required form. - David A. Corneth, Oct 15 2018

David A. Corneth, Table of n, a(n) for n = 1..16289 (First 1316 terms by Marius A. Burtea, terms < 10^11)

Cf. A002383, A002384, A088503, A001358.

[4*p - 3: p in PrimesUpTo(10^5)|IsSquare (4*p - 3)]; // Vincenzo Librandi, Oct 17 2018

Select[ 4Prime[ Range[2000]] - 3, IntegerQ[ Sqrt[ # ]] &] (* Robert G. Wilson v, Sep 20 2005 *)

isok(n) = issquare(n) && (p=(n+3)/4) && (frac(p)==0) && isprime(p); \\ Michel Marcus, Oct 15 2018

upto(n) = my(res = List()); forstep(i = 3, sqrtint(n), 2, if(isprime((i^2+3)/4), listput(res, i^2))); res \\ David A. Corneth, Oct 15 2018

a(n) = 4*A002383(n) - 3 = A088503(n-1)^2.

Extended by Ray Chandler, Sep 07 2005

A217072 Numbers k such that (k^19-1)/(k-1) is prime.

Original entry on oeis.org

2, 10, 11, 12, 14, 19, 24, 40, 45, 46, 48, 65, 66, 67, 75, 85, 90, 103, 105, 117, 119, 137, 147, 164, 167, 179, 181, 205, 220, 235, 242, 253, 254, 263, 268, 277, 303, 315, 332, 337, 366, 369, 370, 389, 399, 404, 424, 431, 446, 449, 480, 481, 506, 509, 521, 523

Offset: 1

Tim Johannes Ohrtmann, Sep 26 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002384, A049409, A100330, A162862, A217070-A217089.

[n: n in [2..1000] |IsPrime((n^19 - 1) div (n - 1))]; // Vincenzo Librandi, Sep 28 2012

Select[Range[2, 1000], PrimeQ[(#^19 - 1)/(# - 1)] &] (* T. D. Noe, Sep 26 2012 *)

is(n)=isprime((n^19-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

More terms from T. D. Noe, Sep 26 2012

A246397 Numbers n such that Phi(12, n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 4, 5, 9, 10, 12, 13, 17, 25, 27, 30, 31, 36, 38, 39, 43, 48, 52, 55, 56, 61, 62, 65, 83, 92, 94, 99, 100, 104, 105, 109, 114, 118, 126, 131, 166, 168, 169, 172, 183, 185, 190, 194, 196, 198, 209, 224, 225, 229, 231, 239, 244, 257, 260, 261, 263, 269, 270, 272, 278, 291, 296, 299, 300, 302, 308, 311

Offset: 1

Eric Chen, Nov 13 2014

nonn

Numbers n such that n^4-n^2+1 is prime, or numbers n such that A060886(n) is prime.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), A246392 (10), A162862 (11), this sequence (12), A217070 (13), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075 (31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078 (43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315 (64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441 (81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442 (243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530 (65536).

Cf. A060886, A125258, A085398, A124990.

A246397:=n->`if`(isprime(n^4-n^2+1),n,NULL): seq(A246397(n),n=1..300); # Wesley Ivan Hurt, Nov 14 2014

Select[Range[350], PrimeQ[Cyclotomic[12, #]] &] (* Vincenzo Librandi, Jan 17 2015 *)

for(n=1,10^3,if(isprime(polcyclo(12,n)),print1(n,", "))); \\ Joerg Arndt, Nov 13 2014

A268043 Numbers k such that k^3 - 1 and k^3 + 1 are both semiprimes.

Original entry on oeis.org

6, 1092, 1932, 2730, 4158, 6552, 11172, 25998, 30492, 55440, 76650, 79632, 85092, 102102, 150990, 152082, 152418, 166782, 211218, 235662, 236208, 248640, 264600, 298410, 300300, 301182, 317772, 380310, 387198, 441798, 476028, 488418

Offset: 1

Vincenzo Librandi, Jan 25 2016

Obviously, k+1 and k-1 are always prime numbers.

If k is a term then m = (k - 1) * (k^2 + k + 1) is a term of A169635, i.e., A001157(m) = A001157(m+2) (De Koninck, 2002). - Amiram Eldar, Apr 19 2024

			a(1) = 6 because 6^3-1 = 215 = 5*43 and 6^3+1 = 217 = 7*31, therefore 215, 217 are both semiprimes.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Jean-Marie De Koninck, On the solutions of sigma_2(n) = sigma_2(n + l), Ann. Univ. Sci. Budapest Sect. Comput. 21 (2002), 127-133.

Subsequence of A014574 and A136242.
Cf. A002384, A055494, A088707, A096173, A096175, A109373.
Cf. A001157, A169635.

IsSemiprime:=func< n | &+[k[2]: k in Factorization(n)] eq 2 >; [ n: n in [2..300000] | IsSemiprime(n^3+1) and IsSemiprime(n^3-1) ];

Select[Range[500000], PrimeOmega[#^3 - 1] == PrimeOmega[#^3 + 1] == 2 &]
Select[Range[10^6], And @@ PrimeQ[{# - 1, # + 1, #^2 - # + 1, #^2 + # + 1}] &] (* Amiram Eldar, Apr 19 2024 *)

isok(n) = (bigomega(n^3-1) == 2) && (bigomega(n^3+1) == 2); \\ Michel Marcus, Jan 26 2016

is(n) = isprime(n - 1) && isprime(n + 1) && isprime(n^2 - n + 1) && isprime(n^2 + n + 1); \\ Amiram Eldar, Apr 19 2024

A217073 Numbers k such that (k^23-1)/(k-1) is prime.

Original entry on oeis.org

10, 40, 82, 113, 127, 141, 170, 257, 275, 287, 295, 315, 344, 373, 442, 468, 609, 634, 646, 663, 671, 710, 819, 834, 857, 884, 894, 904, 992, 997, 1060, 1069, 1077, 1120, 1143, 1190, 1232, 1253, 1261, 1280, 1291, 1347, 1407, 1436, 1448, 1483, 1514, 1570, 1642

Offset: 1

Tim Johannes Ohrtmann, Sep 26 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002384, A049409, A100330, A162862, A217070-A217089.

[n: n in [2..1700] |IsPrime((n^23 - 1) div (n - 1))]; // Vincenzo Librandi, Sep 28 2012

Select[Range[2, 1700], PrimeQ[(#^23 - 1)/(# - 1)] &] (* T. D. Noe, Sep 26 2012 *)

is(n)=isprime((n^23-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

More terms from T. D. Noe, Sep 26 2012

A217074 Numbers k such that (k^29-1)/(k-1) is prime.

Original entry on oeis.org

6, 40, 65, 70, 114, 151, 221, 229, 268, 283, 398, 451, 460, 519, 554, 587, 627, 628, 659, 687, 699, 859, 884, 915, 943, 974, 986, 1101, 1120, 1159, 1176, 1212, 1223, 1297, 1312, 1322, 1337, 1390, 1409, 1415, 1446, 1477, 1508, 1592, 1636, 1691, 1800, 1802, 1803

Offset: 1

Tim Johannes Ohrtmann, Sep 26 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002384, A049409, A100330, A162862, A217070-A217089.

[n: n in [2..2000] |IsPrime((n^29 - 1) div (n - 1))]; // Vincenzo Librandi, Sep 28 2012

Select[Range[2, 2000], PrimeQ[(#^29 - 1)/(# - 1)] &] (* T. D. Noe, Sep 26 2012 *)

is(n)=isprime((n^29-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

More terms from T. D. Noe, Sep 26 2012

A217075 Numbers k such that (k^31-1)/(k-1) is prime.

Original entry on oeis.org

2, 14, 19, 31, 44, 53, 71, 82, 117, 127, 131, 145, 177, 197, 203, 241, 258, 261, 276, 283, 293, 320, 325, 379, 387, 388, 406, 413, 461, 462, 470, 486, 491, 534, 549, 569, 582, 612, 618, 639, 696, 706, 723, 746, 765, 767, 774, 796, 802, 877, 878, 903, 923, 981

Offset: 1

Tim Johannes Ohrtmann, Sep 26 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002384, A049409, A100330, A162862, A217070-A217089.

[n: n in [2..1000] |IsPrime((n^31 - 1) div (n - 1))]; // Vincenzo Librandi, Sep 28 2012

Select[Range[2, 1000], PrimeQ[(#^31 - 1)/(# - 1)] &] (* T. D. Noe, Sep 26 2012 *)

is(n)=isprime((n^31-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

More terms from T. D. Noe, Sep 26 2012

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A185632 Primes of the form n^2 + n + 1 where n is nonprime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A217071 Numbers k such that (k^17-1)/(k-1) is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A110284 Squares of the form 4p - 3, where p is a prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

Extensions

A217072 Numbers k such that (k^19-1)/(k-1) is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A246397 Numbers n such that Phi(12, n) is prime, where Phi is the cyclotomic polynomial.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A268043 Numbers k such that k^3 - 1 and k^3 + 1 are both semiprimes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma