A002384 -id:A002384 - cp's OEIS frontend

A217084 Numbers n such that (n^71-1)/(n-1) is prime.

3, 6, 17, 24, 37, 89, 132, 374, 387, 402, 421, 435, 453, 464, 490, 516, 708, 736, 919, 947, 981, 1033, 1067, 1170, 1195, 1253, 1284, 1349, 1385, 1409, 1479, 1709, 1724, 1726, 1735, 1875, 1950, 1984, 2012, 2070, 2124, 2133, 2161, 2194, 2424, 2432, 2459, 2531, 2552

Offset: 1

Tim Johannes Ohrtmann, Sep 26 2012

nonn

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

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

Select[Range[2, 1000], PrimeQ[(#^71 - 1)/(# - 1)] &] (* T. D. Noe, Sep 26 2012 *)

is(n)=isprime((n^71-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

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

A217085 Numbers n such that (n^73-1)/(n-1) is prime.

Original entry on oeis.org

11, 15, 75, 114, 195, 215, 295, 335, 378, 559, 566, 650, 660, 832, 871, 904, 966, 1021, 1112, 1203, 1334, 1433, 1485, 1724, 1822, 1959, 1998, 2115, 2154, 2432, 2465, 2486, 2544, 2559, 2564, 2575, 2611, 2681, 2705, 2735, 2754, 2806, 2880, 3158, 3222, 3306, 3368

Offset: 1

Tim Johannes Ohrtmann, Sep 26 2012

nonn

G. C. Greubel, Table of n, a(n) for n = 1..5000 (terms 1..500 from Vincenzo Librandi)

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

Select[Range[2, 1000], PrimeQ[(#^73 - 1)/(# - 1)] &] (* T. D. Noe, Sep 26 2012 *)

is(n)=isprime((n^73-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

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

A217086 Numbers n such that (n^79-1)/(n-1) is prime.

Original entry on oeis.org

22, 112, 140, 158, 170, 254, 271, 330, 334, 354, 390, 483, 528, 560, 565, 714, 850, 888, 924, 929, 933, 935, 970, 1019, 1047, 1141, 1266, 1338, 1359, 1376, 1412, 1485, 1504, 1542, 1598, 1607, 1618, 1747, 1773, 1814, 1843, 2087, 2088, 2100, 2167, 2186, 2233, 2311

Offset: 1

Tim Johannes Ohrtmann, Sep 26 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..500

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

Select[Range[2, 1000], PrimeQ[(#^79 - 1)/(# - 1)] &] (* T. D. Noe, Sep 26 2012 *)

is(n)=isprime((n^79-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

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

A217087 Numbers n such that (n^83-1)/(n-1) is prime.

Original entry on oeis.org

41, 146, 386, 593, 667, 688, 906, 927, 930, 1025, 1032, 1111, 1410, 1437, 1638, 1829, 1960, 2045, 2381, 2384, 2618, 2807, 2909, 3059, 3164, 3268, 3370, 3783, 3861, 4043, 4054, 4198, 4284, 4539, 4934, 4968, 4992, 5009, 5047, 5049, 5111, 5217, 5237, 5342, 5367

Offset: 1

Tim Johannes Ohrtmann, Sep 26 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300

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

Select[Range[2, 1000], PrimeQ[(#^83 - 1)/(# - 1)] &] (* T. D. Noe, Sep 26 2012 *)

is(n)=isprime((n^83-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

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

A230252 Number of ways to write n = x + y (x, y > 0) with 2*x + 1, x^2 + x + 1 and y^2 + y + 1 all prime.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 4, 4, 3, 4, 1, 3, 3, 3, 5, 5, 4, 3, 6, 4, 7, 7, 2, 4, 6, 4, 4, 6, 3, 1, 4, 2, 4, 7, 4, 1, 4, 4, 2, 6, 4, 3, 4, 2, 3, 5, 3, 2, 1, 2, 3, 6, 2, 6, 6, 3, 5, 4, 5, 3, 7, 2, 4, 6, 2, 4, 5, 3, 5, 8, 5, 2, 10, 4, 4, 8, 5, 6, 7, 8, 4, 11, 4, 3, 6, 4, 2, 4, 8, 8, 11, 5, 3, 11, 5, 3, 6, 4, 5

Offset: 1

Zhi-Wei Sun, Oct 13 2013

nonn

Conjecture: (i) a(n) > 0 for all n > 1.
(ii) Any integer n > 3 can be written as p + q with p, 2*p - 3 and q^2 + q + 1 all prime. Also, each integer n > 3 not equal to 30 can be expressed as p + q with p, q^2 + q - 1 and q^2 + q + 1 all prime.
(iii) Any integer n > 1 can be written as x + y (x, y > 0) with x^2 + 1 (or 4*x^2+1) and y^2 + y + 1 (or 4*y^2 + 1) both prime.
(iv) Each integer n > 3 can be expressed as p + q (q > 0) with p, 2*p - 3 and 4*q^2 + 1 all prime.
(v) Any even number greater than 4 can be written as p + q with p, q and p^2 + 4 (or p^2 - 2) all prime. Also, each even number greater than 2 and not equal to 122 can be expressed as p + q with p, q and (p-1)^2 + 1 all prime.
We have verified the first part for n up to 10^8.

			a(5) = 2 since 5 = 2 + 3 = 3 + 2, and 2*2+1 = 5, 2*3+1 = 7, 2^2+2+1 = 7, 3^2+3+1 = 13 are all prime.
a(31) = 1 since 31 = 14 + 17, and 2*14+1 = 29, 14^2+14+1 = 211 and 17^2+17+1 = 307 are all prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Cf. A002383, A002384, A002375, A219864, A227908, A227909, A230230, A230241, A230254.

a[n_]:=Sum[If[PrimeQ[2i+1]&&PrimeQ[i^2+i+1]&&PrimeQ[(n-i)^2+n-i+1],1,0],{i,1,n-1}]
Table[a[n],{n,1,100}]

A250177 Numbers n such that Phi_21(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

3, 6, 7, 12, 22, 27, 28, 35, 41, 59, 63, 69, 112, 127, 132, 133, 136, 140, 164, 166, 202, 215, 218, 276, 288, 307, 323, 334, 343, 377, 383, 433, 474, 479, 516, 519, 521, 532, 538, 549, 575, 586, 622, 647, 675, 680, 692, 733, 790, 815, 822, 902, 909, 911, 915, 952, 966, 1025, 1034, 1048, 1093

Offset: 1

Eric Chen, Dec 24 2014

nonn

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), A250392 (10), A162862 (11), A246397 (12), A217070 (13), A250174 (14), A250175 (15), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A250176 (20), this sequence (21), A250178 (22), A217073 (23), A250179 (24), A250180 (25), A250181 (26), A153440 (27), A250182 (28), A217074 (29), A250183 (30), A217075 (31), A006313 (32), A250184 (33), A250185 (34), A250186 (35), A097475 (36), A217076 (37), A250187 (38), A250188 (39), A250189 (40), A217077 (41), A250190 (42), A217078 (43), A250191 (44), A250192 (45), A250193 (46), A217079 (47), A250194 (48), A250195 (49), A250196 (50), 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), A251597 (131072), A244150 (524287), A243959 (1048576).

Cf. A085398 (Least k>1 such that Phi_n(k) is prime).

a250177[n_] := Select[Range[n], PrimeQ@Cyclotomic[21, #] &]; a250177[1100] (* Michael De Vlieger, Dec 25 2014 *)

{is(n)=isprime(polcyclo(21,n))};
for(n=1,100, if(is(n)==1, print1(n, ", "), 0)) \\ G. C. Greubel, Apr 14 2018

A174969 Composites of form n^2 + n + 1.

Original entry on oeis.org

21, 57, 91, 111, 133, 183, 273, 343, 381, 507, 553, 651, 703, 813, 871, 931, 993, 1057, 1191, 1261, 1333, 1407, 1561, 1641, 1807, 1893, 1981, 2071, 2163, 2257, 2353, 2451, 2653, 2757, 2863, 3081, 3193, 3423, 3661, 3783, 4033, 4161, 4291, 4557, 4693, 4971

Offset: 1

Michel Lagneau, Apr 02 2010

nonn

			n=1 gives 1^2+1+1=3, which is prime and so not a term, and similarly for n=2,3; n=4 gives 21=3*7, which is a(1).

L. Poletti, Le serie dei numeri primi appartenente alle due forme quadratiche (A) n^2+n+1 e (B) n^2+n-1 per l'intervallo compreso entro 121 milioni, e cioè per tutti i valori di n fino a 11000, Atti della Reale Accademia Nazionale dei Lincei, Memorie della Classe di Scienze Fisiche, Matematiche e Naturali, s. 6, v. 3 (1929), pages 193-218.

Michel Marcus, Table of n, a(n) for n = 1..10000
C. K. Caldwell, Composite Numbers

Cf. A174967, A002384, A002383, A110284.

with(numtheory):for n from 0 to 200 do:x:=n^2+n+1: if type(x,prime)=false then print (x):else fi:od:

Select[Array[ #^2+#+1&,6!,2],!PrimeQ[ # ]&] (* Vladimir Joseph Stephan Orlovsky, Apr 07 2010 *)

isok(k) = (k>1) && !isprime(k) && issquare(4*k-3); \\ Michel Marcus, Apr 20 2022

Example edited and keyword uned removed by D. S. McNeil, Nov 17 2010

A182253 Nonprime numbers n such that n^2 + n + 1 is prime.

Original entry on oeis.org

1, 6, 8, 12, 14, 15, 20, 21, 24, 27, 33, 38, 50, 54, 57, 62, 66, 69, 75, 77, 78, 80, 90, 99, 105, 110, 111, 117, 119, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 168, 176, 188, 189, 192, 194, 203, 206, 209, 215, 218, 231, 236, 245, 246, 266, 272, 278

Offset: 1

Bernard Schott, Dec 18 2012

nonn

All these numbers are in A002384 but not in A053182.
The generated prime numbers n^2 + n + 1 are in A185632.
All the generated numbers n^2 + n + 1 = 111_n are by definition Brazilian numbers: A125134. See Links: "Les nombres brésiliens" - Section V.5 page 35.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Cf. A002383, A002384, A053182, A053183, A085104, A125134, A185632.

Select[Range@ 280, And[! PrimeQ@ #, PrimeQ[#^2 + # + 1]] &] (* Michael De Vlieger, Jul 30 2017 *)

isok(n) = ! isprime(n) && isprime(n^2 + n + 1); \\ Michel Marcus, Sep 04 2013

A075714 1+n+n^s is a prime, s=18.

Original entry on oeis.org

1, 2, 9, 24, 27, 44, 80, 251, 263, 311, 332, 356, 366, 371, 458, 515, 546, 548, 561, 566, 597, 599, 608, 650, 674, 713, 717, 722, 746, 762, 855, 867, 909, 969, 989, 993, 1010, 1011, 1022, 1052, 1064, 1191, 1245, 1269, 1275, 1284, 1355, 1376, 1431, 1473

Offset: 1

Zak Seidov, Oct 03 2002

For s = 5,8,11,14,17,20,..., n_s=1+n+n^s is always composite for any n>1. Also at n=1, n_s=3 is a prime for any s. So it is interesting to consider only the cases of s =/= 5,8,11,14,17,20,... and n>1. Here i consider the case s=18 and find several first n's making n_s a prime (or a probable prime).

			2 is OK because at s=18, n=2, n_s=1+n+n^s=262147 is a prime.

Vincenzo Librandi, Table of n, a(n) for n = 1..1500

Cf. A002384, A075713, A075715.

[n: n in [0..1600] | IsPrime(s) where s is 1+n+n^18]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1800], PrimeQ[1 + # + #^18] &]  (* Harvey P. Dale, Mar 24 2011 *)

for(n=1,1000,if(isprime(1+n+n^18),print1(n",")))

More terms from Ralf Stephan, Apr 05 2003

A075715 Numbers n such that n^16 + n + 1 is prime.

Original entry on oeis.org

1, 2, 21, 26, 47, 65, 99, 102, 206, 215, 216, 257, 294, 342, 437, 441, 537, 540, 702, 747, 837, 860, 909, 912, 921, 926, 942, 1020, 1071, 1101, 1112, 1125, 1181, 1254, 1266, 1322, 1344, 1364, 1370, 1406, 1422, 1665, 1814, 1821, 1829, 1905, 2024

Offset: 1

Zak Seidov, Oct 03 2002

For s = 5, 8, 11, 14, 17, 20, ..., n_s = 1 + n + n^s is always composite for any n > 1. Also at n = 1, n_s = 3 is a prime for any s. So it is interesting to consider only the cases of s =/= 5, 8, 11, 14, 17, 20, ... and n > 1. Here we consider the case s = 16 and find several first n's making n_s a prime (or a probable prime).

			2 is in the sequence because 1 + 2 + 2^16 = 65539 is prime.

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

Cf. A002384, A075714, A075716.

[n: n in [1..3000]| IsPrime(n^16+n+1)]; // Vincenzo Librandi, Dec 17 2013

A075715:=n->if type(1+n+n^16, prime) then n; fi; seq(A075715(n), n=1..3000); # Wesley Ivan Hurt, Dec 17 2013

Select[Range[3000], PrimeQ[#^16 + # + 1] &] (* Vincenzo Librandi, Dec 17 2013 *)

for(n=1,3000,if(isprime(1+n+n^16),print1(n",")))

More terms from Ralf Stephan, Mar 19 2003

cp's OEIS Frontend

A217084 Numbers n such that (n^71-1)/(n-1) is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A217085 Numbers n such that (n^73-1)/(n-1) is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A217086 Numbers n such that (n^79-1)/(n-1) is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A217087 Numbers n such that (n^83-1)/(n-1) is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A230252 Number of ways to write n = x + y (x, y > 0) with 2*x + 1, x^2 + x + 1 and y^2 + y + 1 all prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A250177 Numbers n such that Phi_21(n) is prime, where Phi is the cyclotomic polynomial.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

A174969 Composites of form n^2 + n + 1.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Extensions

A182253 Nonprime numbers n such that n^2 + n + 1 is prime.

Views

Author

Keywords

Comments

Links

Crossrefs