A246392 -id:A246392 - cp's OEIS frontend

A085398 Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is prime.

3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 6, 2, 4, 3, 2, 10, 2, 22, 2, 2, 4, 6, 2, 2, 2, 2, 2, 14, 3, 61, 2, 10, 2, 14, 2, 15, 25, 11, 2, 5, 5, 2, 6, 30, 11, 24, 7, 7, 2, 5, 7, 19, 3, 2, 2, 3, 30, 2, 9, 46, 85, 2, 3, 3, 3, 11, 16, 59, 7, 2, 2, 22, 2, 21, 61, 41, 7, 2, 2, 8, 5, 2, 2

Offset: 1

Don Reble, Jun 28 2003

nonn

Conjecture: a(n) is defined for all n. - Eric Chen, Nov 14 2014

Existence of a(n) is implied by Bunyakovsky's conjecture. - Robert Israel, Nov 13 2014

			a(11) = 5 because C11(k) is composite for k = 2, 3, 4 and prime for k = 5.
a(37) = 61 because C37(k) is composite for k = 2, 3, 4, ..., 60 and prime for k = 61.

Jinyuan Wang, Table of n, a(n) for n = 1..5000 (terms 1..1500 from Eric Chen)
Wikipedia, Bunyakowsky conjecture

Cf. A117544, A066180, A085399, A103795, A056993, A153438, A246119, A246120, A246121, A206418, A205506, A181980.

Cf. A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A006314, A217071, A164989, A217072, A217073, A153440, A217074, A217075, A006313, A097475.

f:= proc(n) local k;
for k from 2 do if isprime(numtheory:-cyclotomic(n,k)) then return k fi od
end proc:
seq(f(n), n = 1 .. 100); # Robert Israel, Nov 13 2014

Table[k = 2; While[!PrimeQ[Cyclotomic[n, k]], k++]; k, {n, 300}] (* Eric Chen, Nov 14 2014 *)

a(n) = k=2; while(!isprime(polcyclo(n, k)), k++); k; \\ Michel Marcus, Nov 13 2014

a(A072226(n)) = 2. - Eric Chen, Nov 14 2014
a(n) = A117544(n) except when n is a prime power, since if n is a prime power, then A117544(n) = 1. - Eric Chen, Nov 14 2014
a(prime(n)) = A066180(n), a(2*prime(n)) = A103795(n), a(2^n) = A056993(n-1), a(3^n) = A153438(n-1), a(2*3^n) = A246120(n-1), a(3*2^n) = A246119(n-1), a(6^n) = A246121(n-1), a(5^n) = A206418(n-1), a(6*A003586(n)) = A205506(n), a(10*A003592(n)) = A181980(n).

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

Original entry on oeis.org

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

A250185 Numbers n such that Phi(34,n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 7, 13, 17, 19, 41, 48, 58, 59, 66, 86, 129, 133, 139, 143, 146, 149, 166, 167, 231, 268, 270, 299, 328, 359, 387, 397, 408, 469, 523, 527, 534, 541, 553, 555, 569, 582, 583, 600, 608, 634, 664, 667, 672, 673, 709, 714, 720, 725, 733, 746, 759, 776, 802, 808, 822, 860, 870, 877, 892, 896, 902, 911, 962, 970, 975, 1034, 1050, 1051, 1082

Offset: 1

R. J. Mathar, Jan 12 2015

nonn

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

Cf. A246392.

[n: n in [1..2000]| IsPrime((n^17+1) div (n+1))]; // Vincenzo Librandi, Jan 15 2015

Select[Range[2000], PrimeQ[(#^17 + 1) / (# + 1)] &] (*  Vincenzo Librandi, Jan 15 2015 *)

is(n)=isprime(polcyclo(34,n)) \\ Charles R Greathouse IV, Sep 08 2015

A250187 Numbers n such that Phi(38,n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 10, 13, 24, 28, 43, 47, 49, 54, 65, 83, 98, 106, 143, 152, 177, 184, 190, 194, 195, 241, 249, 259, 264, 292, 315, 319, 345, 353, 355, 386, 394, 481, 500, 517, 525, 534, 535, 556, 595, 601, 649, 656, 680, 686, 687, 697, 707, 710, 756, 798, 804, 817, 818, 829, 839, 841, 858, 864, 891, 906, 912, 932, 948, 973, 991, 994, 1012

Offset: 1

R. J. Mathar, Jan 12 2015

nonn

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

Cf. A246392.

[n: n in [1..2000]| IsPrime((n^19+1) div (n+1))]; // Vincenzo Librandi, Jan 15 2015

Select[Range[2000], PrimeQ[(#^19 + 1) / (# + 1)] &] (* Vincenzo Librandi, Jan 15 2015 *)

select(x->isprime(x), vector(1000, n, polcyclo(38, n)), 1) \\ Michel Marcus, Jan 15 2015

A250193 Numbers n such that Phi(46,n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 7, 16, 17, 18, 25, 46, 47, 106, 110, 111, 118, 136, 144, 145, 230, 238, 361, 382, 422, 439, 474, 494, 495, 519, 588, 639, 657, 707, 733, 751, 802, 831, 863, 902, 925, 976, 994, 1001, 1059, 1140, 1181, 1240, 1249, 1319, 1375, 1442, 1458, 1508, 1699, 1734, 1751, 1757, 1760, 1766, 1807, 1849, 1897, 1904, 1914, 1922

Offset: 1

R. J. Mathar, Jan 12 2015

nonn

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

Cf. A246392.

[n: n in [1..2000]| IsPrime((n^23+1) div (n+1))]; // Vincenzo Librandi, Jan 15 2015

Select[Range[2000], PrimeQ[(#^23 + 1) / (# + 1)] &] (* Vincenzo Librandi, Jan 15 2015 *)

is(n)=isprime(polcyclo(46,n)) \\ Charles R Greathouse IV, Sep 08 2015

A060884 a(n) = n^4 - n^3 + n^2 - n + 1.

Original entry on oeis.org

1, 1, 11, 61, 205, 521, 1111, 2101, 3641, 5905, 9091, 13421, 19141, 26521, 35855, 47461, 61681, 78881, 99451, 123805, 152381, 185641, 224071, 268181, 318505, 375601, 440051, 512461, 593461, 683705, 783871, 894661, 1016801, 1151041, 1298155, 1458941, 1634221

Offset: 0

N. J. A. Sloane, May 05 2001

a(n) = Phi_10(n), where Phi_k is the k-th cyclotomic polynomial.
Number of walks of length 5 between any two distinct nodes of the complete graph K_{n+1} (n>=1). Example: a(1)=1 because in the complete graph AB we have only one walk of length 5 between A and B: ABABAB. - Emeric Deutsch, Apr 01 2004
t^4-t^3+t^2-t+1 is the Alexander polynomial (with negative powers cleared) of the cinquefoil knot (torus knot T(5,2)). The associated Seifert matrix S is [[ -1, -1, 0, -1], [ 0, -1, 0, 0], [ -1, -1, -1, -1], [ 0, -1, 0, -1]]. a(n) = det(transpose(S)-n*S). Cf. A084849. - Peter Bala, Mar 14 2012
For odd n, a(n) * (n+1) / 2 also represents the first integer in a sum of n^5 consecutive integers that equals n^10. - Patrick J. McNab, Dec 26 2016

Ray Chandler, Table of n, a(n) for n = 0..10000 (first 1001 terms from Harry J. Smith)
Index to values of cyclotomic polynomials of integer argument
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A084849, A246392, A259257.

A060884 := proc(n)
        numtheory[cyclotomic](10,n) ;
end proc:
seq(A060884(n),n=0..20) ; # R. J. Mathar, Feb 07 2014

Table[1 + Fold[(-1)^(#2)*n^(#2) + #1 &, Range[0, 4]], {n, 0, 33}] (* or *)
CoefficientList[Series[(1 - 4 x + 16 x^2 + 6 x^3 + 5 x^4)/(1 - x)^5, {x, 0, 33}], x] (* Michael De Vlieger, Dec 26 2016 *)
Table[n^4-n^3+n^2-n+1,{n,0,40}] (* or *) LinearRecurrence[{5,-10,10,-5,1},{1,1,11,61,205},40] (* Harvey P. Dale, Sep 08 2018 *)

a(n) = { n^4 - n^3 + n^2 - n + 1 } \\ Harry J. Smith, Jul 13 2009

G.f.: (1-4*x+16*x^2+6*x^3+5*x^4)/(1-x)^5. - Emeric Deutsch, Apr 01 2004

E.g.f.: exp(x)*(1 + 5*x^2 + 5*x^3 + x^4). - Stefano Spezia, Apr 22 2023

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,", ")))

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

A250175 Numbers n such that Phi_15(n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 11, 17, 23, 43, 46, 52, 53, 61, 62, 78, 84, 88, 89, 92, 99, 108, 123, 124, 141, 146, 154, 156, 158, 163, 170, 171, 182, 187, 202, 217, 219, 221, 229, 233, 238, 248, 249, 253, 264, 274, 275, 278, 283, 285, 287, 291, 296, 302, 309, 314, 315, 322, 325, 342, 346, 353, 356, 366, 368, 372, 377, 380, 384, 394, 404, 406, 411, 420, 425

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), A246392 (10), A162862(11), A246397 (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).

Select[Range[600], PrimeQ[Cyclotomic[15, #]] &] (* Vincenzo Librandi, Jan 16 2015 *)

isok(n) = isprime(polcyclo(15, n)); \\ Michel Marcus, Jan 16 2015

More terms from Vincenzo Librandi, Jan 16 2015

cp's OEIS Frontend

A085398 Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is prime.

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

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A250185 Numbers n such that Phi(34,n) is prime, where Phi is the cyclotomic polynomial.

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Magma

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A250187 Numbers n such that Phi(38,n) is prime, where Phi is the cyclotomic polynomial.

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Magma

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A250193 Numbers n such that Phi(46,n) is prime, where Phi is the cyclotomic polynomial.

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A060884 a(n) = n^4 - n^3 + n^2 - n + 1.

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

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Magma

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

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