A006314 -id:A006314 - cp's OEIS frontend

A250176 Numbers n such that Phi_20(n) is prime, where Phi is the cyclotomic polynomial.

4, 9, 11, 16, 19, 26, 34, 45, 54, 70, 86, 91, 96, 101, 105, 109, 110, 119, 120, 126, 129, 139, 141, 149, 171, 181, 190, 195, 215, 229, 260, 276, 299, 305, 309, 311, 314, 319, 334, 339, 369, 375, 414, 420, 425, 444, 470, 479, 485, 506, 519, 534, 540, 550

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[20, #]] &] (* Vincenzo Librandi, Jan 16 2015 *)

isok(n) = isprime(polcyclo(20, n)); \\ Michel Marcus, Sep 29 2015

More terms from Vincenzo Librandi, Jan 16 2015

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)

A258805 Primes of the form k^8 + 1.

Original entry on oeis.org

2, 257, 65537, 37588592026706177, 92170395205042177, 147578905600000001, 284936905588473857, 3503536769037500417, 11007531417600000001, 11763130845074473217, 47330370277129322497, 50024641296100000001, 76872571987558646017, 416806419029812551937

Offset: 1

Vincenzo Librandi, Jun 11 2015

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A002496, A037896.

Cf. A006314 (associated n), A060890.

[a: n in [1..500] | IsPrime(a) where a is n^8+1];

Mathematica
```
Select[Range[500]^8 + 1, PrimeQ]
```

is(n)=ispower(n-1,8) && isprime(n) \\ Charles R Greathouse IV, Jun 11 2015

a(n) = A060890(A006314(n)). - Michel Marcus, Jun 11 2015

A235983 Numbers n of the form p^8 + 1 (for prime p) such that n^8 + 1 is also prime.

Original entry on oeis.org

5764802, 146830437604322, 498311414318121122, 554786279519086052117762, 830149880552636599409282, 12628864335244989661982882, 33144490094099439467757602, 47203563969247823515902242, 179357590196404221918909122, 397370823547272766854136322, 2043714066708245412886790402, 139717795608648816763227344162

Offset: 1

Derek Orr, Jan 17 2014

nonn

All numbers are congruent to 2 mod 20.

			33144490094099439467757602 = 1549^8 + 1 (1549 is prime) and 33144490094099439467757602^8 + 1 is prime, so 33144490094099439467757602 is a member of this sequence.

Cf. A060890, A006314.

Select[Prime[Range[600]]^8+1,PrimeQ[#^8+1]&] (* Harvey P. Dale, Dec 21 2014 *)

import sympy
from sympy import isprime
{print(n**8+1) for n in range(10000) if isprime(n) if isprime((n**8+1)**8+1)}

A087738 Square array: T(n,k) gives n-th number a such that a^(2^k)+1 is prime (a generalized Fermat).

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 6, 4, 2, 1, 10, 6, 4, 2, 1, 12, 10, 6, 4, 2, 1, 16, 14, 16, 118, 44, 30, 1, 18, 16, 20, 132, 74, 54, 102, 1, 22, 20, 24, 140, 76, 96, 162, 120, 1, 28, 24, 28, 152, 94, 112, 274, 190, 278, 1, 30, 26, 34, 208, 156, 114, 300, 234, 614, 46, 1, 36, 36, 46, 240, 158

Offset: 0

Jeppe Stig Nielsen, Oct 01 2003

			{1}; {2,1}; {4,2,1}; ...
See the well-formed array on Gallot's page.

Harvey Dubner, J. Recr. Math., 18, 1986.

Yves Gallot, Generalized Fermat Prime Search.

Cf. A056993, A006093, A005574, A000068, A006314, A006313, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362.

A121978 Numbers k such that (2*k^2)^8 + 1 is prime.

Original entry on oeis.org

1, 11, 12, 14, 20, 27, 29, 30, 40, 65, 71, 85, 87, 89, 97, 104, 115, 147, 155, 175, 181, 189, 194, 244, 263, 264, 285, 286, 291, 303, 354, 360, 370, 376, 403, 407, 417, 423, 429, 433, 448, 479, 492, 493, 520, 570, 582, 588, 596, 617, 627, 629, 644, 654, 661

Offset: 1

Alexander Adamchuk, Sep 10 2006

Corresponding primes of the form (2*k^2)^8 + 1 are {257, 11763130845074473217, 47330370277129322497, 557556054479199010817, 167772160000000000000001, ...}.

There are consecutive twin pairs {a(n),a(n+1)} = {11,12}, {29,30}, {263,264},{285,286}, {492,493}, {833,834}, ...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A006314.

Select[Range[1000],PrimeQ[(2*#1^2)^8+1]&]
Select[Range[1000],PrimeQ[256#^16+1]&] (* Harvey P. Dale, Nov 04 2020 *)

is(n)=isprime((2*n^2)^8+1) \\ Charles R Greathouse IV, Jun 13 2017

A217993 Smallest k such that k^(2^n) + 1 and (k+2)^(2^n) + 1 are both prime.

Original entry on oeis.org

2, 2, 2, 2, 74, 112, 2162, 63738, 13220, 54808, 3656570, 6992032, 125440, 103859114, 56414914, 87888966

Offset: 0

Michel Lagneau, Oct 17 2012

a(15)=87888966 but a(14) is unknown. - Jeppe Stig Nielsen, Mar 17 2018

The prime pair related to a(14) was found four days ago, and today double checking has proved that they are indeed the first occurrence for n=14. - Jeppe Stig Nielsen, May 02 2018

			a(0) = 2 because 2^1+1 = 3 and 4^1+1 = 5 are prime;
a(1) = 2 because 2^2+1 = 5  and 4^2+1 = 17 are prime;
a(2) = 2 because 2^4+1 = 17  and 4^4+1 = 257 are prime;
a(3) = 2 because  2^8+1 = 257 and 4^8+1 = 65537 are prime.

Cf. A006313, A006314, A006315, A006316, A056994, A056995, A057465, A057002, A088361, A088362, A118539.

for n from 0  to 5 do:ii:=0:for k from 2 by 2 to 10000 while(ii=0) do:if type(k^(2^n)+1,prime)=true and type((k+2)^(2^n)+1,prime)=true then ii:=1: printf ( "%d %d \n",n,k):else fi:od:od:

a(n) = A118539(n)-1. - Jeppe Stig Nielsen, Feb 27 2016

a(13) from Jeppe Stig Nielsen, Mar 17 2018

a(14) and a(15) from Jeppe Stig Nielsen, May 02 2018

A242553 Least number k such that n^8 + k^8 is prime.

Original entry on oeis.org

1, 1, 10, 1, 6, 5, 12, 13, 16, 3, 24, 7, 2, 3, 8, 9, 4, 17, 4, 7, 2, 3, 20, 7, 8, 19, 10, 3, 10, 19, 14, 17, 32, 11, 8, 25, 6, 25, 40, 7, 10, 43, 16, 5, 68, 7, 30, 5, 8, 19, 58, 17, 26, 17, 2, 11, 10, 3, 4, 49, 6, 71, 22, 15, 14, 47, 30, 9, 2, 19, 6, 19, 6, 5, 28, 13, 2

Offset: 1

Derek Orr, May 17 2014

nonn

If a(n) = 1, then n is in A006314.

			10^8+1^8 = 100000001 is not prime. 10^8+2^8 = 100000256 is not prime. 10^8+3^8 = 100006561 is prime. Thus, a(10) = 3.

Cf. A069003, A006314.

lnk[n_]:=Module[{c=n^8,k=1},While[CompositeQ[c+k^8],k++];k]; Array[lnk,80] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jan 12 2020 *)

a(n)=for(k=1,oo,if(ispseudoprime(n^8+k^8),return(k)));

import sympy
from sympy import isprime
def a(n):
  for k in range(10**4):
    if isprime(n**8+k**8):
      return k
n = 1
while n < 100:
  print(a(n))
  n += 1

A272137 Primes of the form k^16 + 1.

Original entry on oeis.org

2, 65537, 197352587024076973231046657, 808551180810136214718004658177, 1238846438084943599707227160577, 37157429083410091685945089785857, 123025056645280288014028950372089857, 150838912030874130174020868290707457

Offset: 1

Jaroslav Krizek, May 08 2016

nonn

Corresponding values of k are in A006313.

Jaroslav Krizek, Table of n, a(n) for n = 1..100

Cf. Sequences of numbers n such that n^(2^k)+1 is a prime p for k = 1-13: A005574 (k=1), A000068 (k=2), A006314 (k=3), A006313 (k=4), A006315 (k=5), A006316 (k=6), A056994 (k=7), A056995 (k=8), A057465 (k=9), A057002 (k=10), A088361 (k=11), A088362 (k=12), A226528 (k=13).

Corresponding sequences of primes p of the form n^(2^k)+1 for k = 1-4: A002496 (k=1), A037896 (k=2), A258805 (k=3), A272137 (k=4).

[n^16 + 1: n in [1..700] | IsPrime(n^16 + 1)];

A272137:=n->`if`(isprime(n^16+1), n^16+1, NULL): seq(A272137(n), n=1..200); # Wesley Ivan Hurt, May 11 2016

A217972 Numbers n such that n^8 + 1 and (n + 2)^8 + 1 are both prime.

Original entry on oeis.org

2, 240, 288, 508, 540, 680, 916, 1614, 2328, 2872, 2960, 2988, 3402, 3708, 3770, 4760, 4762, 4810, 5370, 5490, 5776, 5878, 6204, 7276, 7890, 8414, 8652, 9418, 9858, 11218, 11896, 12510, 13328, 13938, 14418, 15846, 16422, 17206, 18152, 18954, 19226, 20640

Offset: 1

Michel Lagneau, Oct 17 2012

			2 is in the sequence because 2^8 + 1 = 257 and 4^8 + 1 = 65537 are both prime.
4 is not in the sequence because although 4^8 + 1 is a prime (as we saw above), 6^8 + 1 is not, being a multiple of 17.

Cf. A006314.

lst = {}; Do[p = n^8 + 1; q = (n + 2)^8 + 1; If[PrimeQ[p] && PrimeQ[q], AppendTo[lst, n]], {n, 0, 21000}]; lst (* Lagneau *)
Select[Range[10^5], PrimeQ[#^8 + 1] && PrimeQ[(# + 2)^8 + 1] &] (* Alonso del Arte, Oct 17 2012 *)

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A250176 Numbers n such that Phi_20(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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A258805 Primes of the form k^8 + 1.

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A235983 Numbers n of the form p^8 + 1 (for prime p) such that n^8 + 1 is also prime.

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A087738 Square array: T(n,k) gives n-th number a such that a^(2^k)+1 is prime (a generalized Fermat).

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A121978 Numbers k such that (2*k^2)^8 + 1 is prime.

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A217993 Smallest k such that k^(2^n) + 1 and (k+2)^(2^n) + 1 are both prime.

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Maple

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A242553 Least number k such that n^8 + k^8 is prime.

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