A250174 -id:A250174 - cp's OEIS frontend

A100330 Positive integers k such that k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 is prime.

1, 2, 3, 5, 6, 13, 14, 17, 26, 31, 38, 40, 46, 56, 60, 61, 66, 68, 72, 73, 80, 87, 89, 93, 95, 115, 122, 126, 128, 146, 149, 156, 158, 160, 163, 180, 186, 192, 203, 206, 208, 220, 221, 235, 237, 238, 251, 264, 266, 280, 282, 290, 294, 300, 303, 320, 341, 349, 350

Offset: 1

Ray G. Opao, Nov 16 2004

nonn

The corresponding primes are A088550. - Bernard Schott, Dec 20 2012

k = 5978493 * 2^150006 - 1 is an example of a very large term of this sequence. The generated prime is proved by the N-1 method (because k is prime and k*(k+1) is fully factored and this provides for an exactly 33.33...% factorization for Phi_7(k) - 1). - Serge Batalov, Mar 13 2015

			2 is in the sequence because 2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2 + 1 = 127, which is prime.

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

Cf. A100331, A250174 (Phi_14(n) = n^6 - n^5 + n^4 - n^3 + n^2 - n + 1 primes; these two sequences can also be considered an extension of each other into negative n values), A250177 (Phi_21(n) primes).

[n: n in [1..500]| IsPrime(n^6 + n^5 + n^4 + n^3 + n^2 + n + 1)]; // Vincenzo Librandi, Feb 08 2014

A100330 := proc(n)
    option remember;
    local a;
    if n = 1 then
        1;
    else
        for a from procname(n-1)+1 do
            if isprime(numtheory[cyclotomic](7,a)) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A100330(n),n=1..30) ; # R. J. Mathar, Feb 07 2014

Select[Range[350], PrimeQ[Sum[ #^i, {i, 0, 6}]] &] (* Ray Chandler, Nov 17 2004 *)
Do[If[PrimeQ[n^6 + n^5 + n^4 + n^3 + n^2 + n + 1], Print[n]], {n, 1, 500}] (* Vincenzo Librandi, Feb 08 2014 *)

is(n)=isprime(polcyclo(7,n)) \\ Charles R Greathouse IV, Apr 28 2015

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

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

A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

Original entry on oeis.org

1, 1, -1, 1, 0, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 3, 4, 7, 2, 1, 1, 4, 5, 13, 5, 5, 1, 1, 5, 6, 21, 10, 31, 1, 1, 1, 6, 7, 31, 17, 121, 3, 7, 1, 1, 7, 8, 43, 26, 341, 7, 127, 2, 1, 1, 8, 9, 57, 37, 781, 13, 1093, 17, 3, 1, 1, 9, 10, 73, 50, 1555, 21, 5461, 82, 73, 1, 1, 1, 10, 11, 91, 65, 2801, 31, 19531, 257, 757, 11, 11, 1, 1, 11, 12, 111, 82, 4681, 43, 55987, 626, 4161, 61, 2047, 1, 1

Offset: 0

Eric Chen, Apr 22 2015

Outside of rows 0, 1, 2 and columns 0, 1, only terms of A206942 occur.
Conjecture: There are infinitely many primes in every row (except row 0) and every column (except column 0), the indices of the first prime in n-th row and n-th column are listed in A117544 and A117545. (See A206864 for all the primes apart from row 0, 1, 2 and column 0, 1.)
Another conjecture: Except row 0, 1, 2 and column 0, 1, the only perfect powers in this table are 121 (=Phi_5(3)) and 343 (=Phi_3(18)=Phi_6(19)).

			Read by antidiagonals:
m\n  0   1   2   3   4   5   6   7   8   9  10  11  12
------------------------------------------------------
0    1   1   1   1   1   1   1   1   1   1   1   1   1
1   -1   0   1   2   3   4   5   6   7   8   9  10  11
2    1   2   3   4   5   6   7   8   9  10  11  12  13
3    1   3   7  13  21  31  43  57  73  91 111 133 157
4    1   2   5  10  17  26  37  50  65  82 101 122 145
5    1   5  31 121 341 781 ... ... ... ... ... ... ...
6    1   1   3   7  13  21  31  43  57  73  91 111 133
etc.
The cyclotomic polynomials are:
n        n-th cyclotomic polynomial
0        1
1        x-1
2        x+1
3        x^2+x+1
4        x^2+1
5        x^4+x^3+x^2+x+1
6        x^2-x+1
...

Eric Chen, Table of n, a(n) for n = 0..5049 (first 100 antidiagonals)
Eric Weisstein's World of Mathematics, Cyclotomic polynomial
Wikipedia, Cyclotomic polynomial
Index entries for Cyclotomic polynomials, values at X

Rows 0-16 are A000012, A023443, A000027, A002061, A002522, A053699, A002061, A053716, A002523, A060883, A060884, A060885, A060886, A060887, A060888, A060889, A060890.
Columns 0-13 are A158388, A020500, A019320, A019321, A019322, A019323, A019324, A019325, A019326, A019327, A019328, A019329, A019330, A019331.
Main diagonal is A070518.
Indices of primes in n-th row for n = 1-20 are A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A250174, A250175, A006314, A217071, A164989, A217072, A250176.
Indices of primes in n-th column for n = 1-10 are A246655, A072226, A138933, A138934, A138935, A138936, A138937, A138938, A138939, A138940.
Indices of primes in main diagonal is A070519.
Cf. A117544 (indices of first prime in n-th row), A085398 (indices of first prime in n-th row apart from column 1), A117545 (indices of first prime in n-th column).
Cf. A206942 (all terms (sorted) for rows>2 and columns>1).
Cf. A206864 (all primes (sorted) for rows>2 and columns>1).

Table[Cyclotomic[m, k-m], {k, 0, 49}, {m, 0, k}]

t1(n)=n-binomial(floor(1/2+sqrt(2+2*n)), 2)
t2(n)=binomial(floor(3/2+sqrt(2+2*n)), 2)-(n+1)
T(m, n) = if(m==0, 1, polcyclo(m, n))
a(n) = T(t1(n), t2(n))

T(m, n) = Phi_m(n)

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,", ")))

cp's OEIS Frontend

A100330 Positive integers k such that k^6 + k^5 + k^4 + k^3 + k^2 + k + 1 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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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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A253240 Square array read by antidiagonals: T(m, n) = Phi_m(n), the m-th cyclotomic polynomial at x=n.

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

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

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