A250178 -id:A250178 - cp's OEIS frontend

A084742 Least k such that (n^k+1)/(n+1) is prime, or 0 if no such prime exists.

3, 3, 3, 5, 3, 3, 0, 3, 5, 5, 5, 3, 7, 3, 3, 7, 3, 17, 5, 3, 3, 11, 7, 3, 11, 0, 3, 7, 139, 109, 0, 5, 3, 11, 31, 5, 5, 3, 53, 17, 3, 5, 7, 103, 7, 5, 5, 7, 1153, 3, 7, 21943, 7, 3, 37, 53, 3, 17, 3, 7, 11, 3, 0, 19, 7, 3, 757, 11, 3, 5, 3, 7, 13, 5, 3, 37, 3, 3, 5, 3, 293, 19, 7, 167, 7, 7, 709, 13, 3, 3, 37, 89, 71, 43, 37

Offset: 2

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 15 2003

nonn

When (n^k+1)/(n+1) is prime, k must be prime. As mentioned by Dubner and Granlund, when n is a perfect power (the power is greater than 2), then (n^k+1)/(n+1) will usually be composite for all k, which is the case for n = 8, 27, 32 and 64. a(n) are only probable primes for n = {53, 124, 150, 182, 205, 222, 296}.
a(n) = 0 if n = {8, 27, 32, 64, 125, 243, ...}. - Eric Chen, Nov 18 2014
More terms: a(124) = 16427, a(150) = 6883, a(182) = 1487, a(205) = 5449, a(222) = 1657, a(296) = 1303. For n up to 300, a(n) is currently unknown only for n = {97, 103, 113, 175, 186, 187, 188, 220, 284}. All other terms up to a(300) are less than 1000. - Eric Chen, Nov 18 2014
a(97) > 31000. - Eric Chen, Nov 18 2014
a(311) = 2707, a(313) = 4451. - Eric Chen, Nov 20 2014
a(n)=3 if and only if n^2-n+1 is a prime; that is, n belongs to A055494. - Thomas Ordowski, Sep 19 2015
From Altug Alkan, Sep 29 2015: (Start)
a(n)=5 if and only if Phi(10, n) is prime and Phi(6, n) is composite. n belongs to A246392.
a(n)=7 if and only if Phi(14, n) is prime, and Phi(10, n) and Phi(6, n) are both composite. n belongs to A250174.
a(n)=11 if and only if Phi(22, n) is prime, and Phi(14, n), Phi(10, n) and Phi(6, n) are all composite. n belongs to A250178.
Where Phi(k, n) is the k-th cyclotomic polynomial. (End)
a(97) > 800000 (or a(97) = 0). - Wang Runsen, May 10 2023

			a(5) = 5 as (5^5 + 1)/(5 + 1) = 1 - 5 + 5^2 - 5^3 + 5^4 = 521 is a prime.
a(7) = 3 as (7^3 + 1)/(7 + 1) = 1 - 7 + 7^2 = 43 is a prime.

Eric Chen, Table of known a(n) up to a(300)
H. Dubner and T. Granlund, Primes of the Form (b^n+1)/(b+1), J. Integer Sequences, 3 (2000), #P00.2.7.
Eric Weisstein's World of Mathematics, Repunit
Robert G. Wilson v, Letter to N. J. A. Sloane, circa 1991.

Cf. A084741, A065507, A084740, A128164, A126659, A103795.

a(n) = {l=List([8, 27, 32, 64, 125, 243, 324, 343]); for(q=1, #l, if(n==l[q], return(0))); k=2; while(k, s=(n^prime(k)+1)/(n+1); if(ispseudoprime(s), return(prime(k))); k++)}
n=2; while(n<361, print1(a(n), ", "); n++) \\ Eric Chen, Nov 25 2014

More terms from T. D. Noe, Jan 22 2004

A260558 Numbers k such that (k^29+1)/(k+1) is prime.

Original entry on oeis.org

7, 15, 25, 62, 119, 123, 154, 245, 285, 294, 295, 357, 371, 476, 626, 664, 690, 708, 723, 737, 768, 783, 803, 825, 826, 835, 841, 842, 867, 871, 897, 904, 934, 953, 1066, 1069, 1088, 1097, 1108, 1183, 1197, 1202, 1259, 1302, 1364, 1461, 1497, 1528, 1559, 1638

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260559-A260573.

[n: n in [1..10000] |IsPrime((n^29 + 1) div (n + 1))];

Select[Range[1, 10000], PrimeQ[(#^29 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^29+1)/(n+1)), print1(n,", ")))

A260573 Numbers n such that (n^97+1)/(n+1) is prime.

Original entry on oeis.org

70, 121, 300, 317, 348, 404, 412, 460, 515, 605, 839, 843, 904, 953, 1130, 1148, 1342, 1466, 1674, 1779, 1855, 2080, 2108, 2193, 2466, 2519, 2597, 2633, 2697, 2756, 2793, 2799, 2846, 2877, 2899, 2929, 2952, 3081, 3244, 3283, 3300, 3315, 3636, 3730, 3739, 3833

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260558-A260572.

[n: n in [1..10000] |IsPrime((n^97 + 1) div (n + 1))]

Select[Range[1, 10000], PrimeQ[(#^97 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^97+1)/(n+1)), print1(n,", ")))

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

A260559 Numbers k such that (k^31+1)/(k+1) is prime.

Original entry on oeis.org

2, 6, 10, 36, 65, 74, 78, 83, 106, 115, 120, 148, 161, 163, 168, 176, 189, 194, 197, 266, 270, 288, 331, 385, 399, 407, 410, 412, 413, 431, 468, 513, 524, 546, 569, 572, 578, 581, 600, 611, 625, 626, 647, 719, 723, 756, 832, 834, 849, 922, 986, 1006, 1007, 1047

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260558, A260560-A260573.

[n: n in [1..10000] |IsPrime((n^31 + 1) div (n + 1))];

Select[Range[1, 10000], PrimeQ[(#^31 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^31+1)/(n+1)), print1(n,", ")))

A260560 Numbers n such that (n^37+1)/(n+1) is prime.

Original entry on oeis.org

16, 19, 21, 49, 56, 63, 71, 74, 77, 83, 92, 96, 99, 160, 172, 197, 198, 230, 241, 280, 283, 415, 425, 448, 490, 520, 627, 691, 735, 784, 803, 829, 842, 853, 871, 872, 893, 894, 973, 981, 989, 1043, 1060, 1061, 1071, 1179, 1182, 1203, 1290, 1299, 1317, 1370, 1389

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260558-A260573.

[n: n in [1..10000] |IsPrime((n^37 + 1) div (n + 1))]

Select[Range[1, 10000], PrimeQ[(#^37 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^37+1)/(n+1)), print1(n,", ")))

A260571 Numbers n such that (n^83+1)/(n+1) is prime.

Original entry on oeis.org

49, 75, 458, 471, 634, 734, 798, 809, 932, 1139, 1268, 1400, 1498, 1963, 1989, 2112, 2177, 2233, 2252, 2349, 2365, 2446, 2729, 2841, 2861, 2887, 3013, 3048, 3239, 3262, 3403, 3464, 3703, 3855, 3883, 4534, 5147, 5189, 5523, 5611, 5778, 6041, 6200, 6336, 6682, 7068

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260558-A260573.

[n: n in [1..10000] |IsPrime((n^83 + 1) div (n + 1))]

Select[Range[1, 10000], PrimeQ[(#^83 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^83+1)/(n+1)), print1(n,", ")))

A260572 Numbers n such that (n^89+1)/(n+1) is prime.

Original entry on oeis.org

16, 20, 93, 195, 227, 325, 465, 758, 888, 911, 1075, 1301, 1590, 1640, 1783, 1807, 2168, 2204, 2231, 2376, 2528, 2591, 2627, 2648, 2909, 2959, 3063, 3109, 3650, 3688, 3709, 3784, 3910, 3943, 4132, 4162, 4385, 4417, 4443, 4613, 5183, 5465, 5574, 5750, 5854, 5975

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260558-A260571, A260573.

[n: n in [1..10000] |IsPrime((n^89 + 1) div (n + 1))]

Select[Range[1, 10000], PrimeQ[(#^89 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^89+1)/(n+1)), print1(n,", ")))

A260561 Numbers n such that (n^41+1)/(n+1) is prime.

Original entry on oeis.org

61, 63, 99, 144, 230, 312, 360, 401, 413, 424, 451, 515, 542, 567, 610, 618, 622, 651, 690, 732, 817, 871, 1007, 1100, 1156, 1278, 1403, 1427, 1460, 1535, 1572, 1604, 1681, 1742, 1802, 1820, 1847, 1903, 1910, 1913, 1978, 2019, 2104, 2134, 2152, 2169, 2309, 2383

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260558-A260573.

[n: n in [1..10000] |IsPrime((n^41 + 1) div (n + 1))]

Select[Range[1, 10000], PrimeQ[(#^41 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^41+1)/(n+1)), print1(n,", ")))

A260562 Numbers n such that (n^43+1)/(n+1) is prime.

Original entry on oeis.org

2, 3, 6, 22, 59, 83, 91, 95, 120, 148, 195, 196, 201, 247, 252, 264, 315, 360, 378, 458, 555, 680, 792, 893, 1025, 1088, 1158, 1171, 1240, 1280, 1416, 1437, 1632, 1661, 1677, 1681, 1849, 1946, 1960, 2007, 2090, 2092, 2225, 2242, 2244, 2377, 2483, 2547, 2596, 2641

Offset: 1

Tim Johannes Ohrtmann, Jul 29 2015

nonn

Tim Johannes Ohrtmann, Table of n, a(n) for n = 1..10000

Cf. A055494, A246392, A250174, A250178, A250181, A250185, A250187, A250193, A260558-A260573.

[n: n in [1..10000] |IsPrime((n^43 + 1) div (n + 1))]

Select[Range[1, 10000], PrimeQ[(#^43 + 1)/(# + 1)] &]

for(n=1,10000, if(isprime((n^43+1)/(n+1)), print1(n,", ")))

cp's OEIS Frontend

A084742 Least k such that (n^k+1)/(n+1) is prime, or 0 if no such prime exists.

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A260558 Numbers k such that (k^29+1)/(k+1) is prime.

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A260573 Numbers n such that (n^97+1)/(n+1) is prime.

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

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A260559 Numbers k such that (k^31+1)/(k+1) is prime.

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Magma

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A260560 Numbers n such that (n^37+1)/(n+1) is prime.

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A260571 Numbers n such that (n^83+1)/(n+1) is prime.

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A260572 Numbers n such that (n^89+1)/(n+1) is prime.

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