A153438 -id:A153438 - 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).

A153439 Numbers k such that k^6 + k^3 + 1 is prime.

Original entry on oeis.org

1, 2, 3, 8, 11, 20, 21, 26, 30, 50, 51, 56, 60, 78, 98, 102, 117, 129, 134, 146, 159, 171, 186, 189, 191, 198, 200, 209, 210, 212, 222, 240, 249, 267, 269, 278, 279, 299, 300, 333, 344, 363, 383, 390, 398, 399, 425, 429, 438, 444, 450, 458, 504, 509, 533, 540

Offset: 1

Pierre CAMI, Dec 26 2008

nonn

n^6+n^3+1 is the cyclotomic polynomial Phi(9,n). Damianou, p. 4, claims that there is strong experimental evidence that k^6 + k^3 + 1 is prime for infinitely many values of k. - Jonathan Vos Post, Jan 06 2011 [Comment corrected by Jon E. Schoenfield, Jan 22 2018]

Pierre CAMI, Table of n, a(n) for n = 1..37835
Pantelis A. Damianou, On prime values of cyclotomic polynomials, arXiv:1101.1152 [math.NT], 2011.

Cf. A153438, A164989.

[n: n in [0..500] | IsPrime(n^3*(n^3+1)+1)] // Vincenzo Librandi, Nov 26 2010

Select[Range@ 600, PrimeQ[#^6 + #^3 + 1] &] (* Michael De Vlieger, Jan 22 2018 *)

is(n)=prime(n^6+n^3+1) \\ Charles R Greathouse IV, Feb 17 2017

Some unclear comments deleted by N. J. A. Sloane, Sep 06 2009

More terms from Vincenzo Librandi, Mar 25 2010

A153440 Numbers k such that k^9*(k^9+1)+1 is prime.

Original entry on oeis.org

1, 2, 11, 44, 45, 56, 62, 63, 110, 170, 219, 234, 245, 261, 263, 333, 395, 398, 402, 413, 428, 434, 437, 498, 557, 558, 578, 633, 692, 695, 723, 731, 750, 761, 774, 794, 797, 804, 806, 846, 854, 855, 863, 906, 923, 926, 977, 1046, 1085, 1086

Offset: 1

Pierre CAMI, Dec 26 2008

nonn

It seems numbers of the form k^n*(k^n+1)+1 with n > 0, k > 1 may be primes only if n has the form 3^j. When n is even, k^(4*n)+k^(2*n)+1=(k^(2*n)+1)^2-(k^n)^2=(k^(2*n)+k^n+1)*(k^(2*n)-k^n+1) so composite. But why if n odd > 3 and not a power of 3, k^n*(k^n+1)+1 is always composite?

Pierre CAMI, Table of n, a(n) for n=1,...,38019

Cf. A153438.

[n: n in [0..1100] | IsPrime(n^9*(n^9+1)+1)]; // Vincenzo Librandi, Jan 17 2015

k9pQ[n_]:=Module[{c=n^9},PrimeQ[c(c+1)+1]]; Select[Range[1200],k9pQ] (* Harvey P. Dale, Oct 29 2014 *)
Select[Range[1100], PrimeQ[(#^9 (#^9 + 1)) + 1] &] (* Vincenzo Librandi, Jan 17 2015 *)

A205506 Least positive integer m > 1 such that 1 - m^k + m^(2*k) is prime, where k=A003586(n).

Original entry on oeis.org

2, 2, 6, 2, 3, 5, 7, 3, 4, 3, 6, 93, 2, 88, 5, 33, 5, 196, 15, 106, 174, 196, 14, 342, 207, 28, 372, 14, 47, 25, 569, 646, 141, 129, 278, 5, 421, 224, 629, 26, 424, 1081, 688, 246, 736, 4392, 124, 484, 759, 791, 4401, 863, 2854, 410, 1044, 22, 848, 1402, 2006

Offset: 1

Lei Zhou, Feb 01 2012

1 - m^k + m^(2*k) equals Phi(6*k,m) when k=2^p*3^q, p>=0, q>=0, which may be prime numbers for certain positive integer m>1.
The Mathematica program given here generates the first 33 terms.  Further terms were generated by OpenPFGW.
a(62)=7426, while A003586(62)=3^8=6561.

			n=1, A003586(1)=1, when m=2, 1-2^1+2^2=3 is prime, so a(1)=2;
n=2, A003586(2)=2, when m=2, 1-2^2+2^4=13 is prime, so a(2)=2;
...
n=7, A003586(7)=9, when m=7, 1-7^9+7^18=1628413557556843 is prime, so a(7)=7.

Cf. A003586, A033845, A056993, A085398, A153438.

fQ[n_] := n == 3 EulerPhi@n; a = Select[6 Range@500, fQ]/6; l =
Length[a]; Table[m = a[[j]]; i = 1;
While[i++; cp = 1 - i^m + i^(2*m); ! PrimeQ[cp]]; i, {j, 1, l}]

from itertools import count
from sympy import isprime, integer_log
def A205506(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return n+x-sum((x//3**i).bit_length() for i in range(integer_log(x,3)[0]+1))
    k = bisection(f,n,n)
    return next(filter(lambda m:isprime(1-m**k+m**(k<<1)),count(2))) # Chai Wah Wu, Oct 22 2024

a(n) = A085398(6*A003586(n)). - Jinyuan Wang, Jan 01 2023

a(n) is smallest positive m such that Phi(A033845(n),m) is prime. - Chai Wah Wu, Sep 16 2024

A153436 a(n) is the least k such that k^n*(k^n+1)-1 is prime.

Original entry on oeis.org

2, 2, 2, 2, 4, 2, 10, 4, 3, 2, 6, 3, 4, 21, 12, 2, 30, 3, 5, 47, 31, 34, 27, 2, 31, 2, 50, 5, 20, 23, 89, 29, 179, 11, 2, 16, 41, 96, 75, 22, 55, 10, 209, 46, 38, 17, 34, 7, 61, 121, 119, 2, 68, 148, 2, 164, 216, 65, 527, 11, 175, 354, 442, 201, 54, 230, 97, 433, 3

Offset: 1

Pierre CAMI, Dec 26 2008

nonn

Pierre CAMI, Table of n, a(n) for n = 1..500

Cf. A101406, A153436, A153438.

a(n) = {my(k=1); while (! isprime(k^n*(k^n+1)-1), k++); k;}

Name corrected by Michel Marcus, Sep 24 2019

A153441 Numbers k such that k^27*(k^27+1)+1 is prime.

Original entry on oeis.org

1, 21, 50, 77, 153, 191, 194, 311, 405, 440, 462, 557, 638, 659, 690, 746, 852, 887, 944, 945, 1140, 1326, 1344, 1452, 1463, 1607, 1632, 1652, 1659, 1683, 1710, 1788, 1812, 1851, 1925, 1943, 1992, 2157, 2294, 2309, 2352, 2402, 2621, 2687, 2700, 2733, 2756

Offset: 1

Pierre CAMI, Dec 26 2008

nonn

It seems numbers of the form k^n*(k^n+1)+1 with n > 0, k > 1 may be primes only if n has the form 3^j. When n is even, k^(4*n)+k^(2*n)+1=(k^(2*n)+1)^2-(k^n)^2=(k^(2*n)+k^n+1)*(k^(2*n)-k^n+1) so composite. But why if n odd > 3 and not a power of 3, k^n*(k^n+1)+1 is always composite ??

Pierre CAMI, Table of n, a(n) for n=1,...,37719

Cf. A153438.

isok(k)  = isprime(k^27*(k^27+1)+1); \\ Michel Marcus, Sep 20 2019

A153442 Numbers k such that k^81*(k^81+1)+1 is prime.

Original entry on oeis.org

1, 209, 210, 842, 1176, 1358, 1370, 1608, 1707, 1845, 1850, 2594, 2880, 2882, 3123, 3384, 4085, 4457, 4469, 4808, 5090, 5186, 5516, 5529, 5867, 5991, 6123, 6144, 6606, 6906, 7001, 7019, 7119, 7430, 7541, 7719, 8031, 8463, 8471, 8486, 8595, 8609, 8627

Offset: 1

Pierre CAMI, Dec 26 2008

nonn

It seems numbers of the form k^n*(k^n+1)+1 with n > 0, k > 1 may be primes only if n has the form 3^j. When n is even, k^(4*n)+k^(2*n)+1=(k^(2*n)+1)^2-(k^n)^2=(k^(2*n)+k^n+1)*(k^(2*n)-k^n+1) so composite. But why if n odd > 3 and not a power of 3, k^n*(k^n+1)+1 is always composite?

Pierre CAMI, Table of n, a(n) for n=1,...,9439

Cf. A153438.

k81Q[k_]:=Module[{k81=k^81},PrimeQ[k81(k81+1)+1]]; Select[Range[9000], k81Q] (* Harvey P. Dale, Aug 28 2011 *)
Select[Range[9000], PrimeQ[(#^81 (#^81 + 1)) + 1] &] (* Vincenzo Librandi, Jan 17 2015 *)

A246119 a(n) is the least k such that k^(2^n)*(k^(2^n) - 1) + 1 is prime.

Original entry on oeis.org

2, 2, 2, 5, 4, 2, 5, 196, 14, 129, 424, 484, 22, 5164, 7726, 13325, 96873, 192098, 712012, 123447

Offset: 0

Serge Batalov, Aug 14 2014

Numbers of the form k^m*(k^m-1)+1 with m > 0, k > 1 may be primes only if m is 3-smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors >3. This series is a subset of A205506 with only m=2^n.
Trivially, a(n) <= a(n+1)^2. This upper bound, indeed, holds for a(4) = a(5)^2, a(7) = a(8)^2 and a(11) = a(12)^2.
The numbers of this form are Generalized Unique primes (see Links section).
a(16)=96873 corresponds to a prime with 653552 decimal digits.
The search for a(17) which corresponds to a 1385044-decimal digit prime was performed on a small Amazon EC2 cloud farm (40 GRID K520 GPUs), at a cost of approximately $1000 over three weeks.
a(18) <= 712012 corresponds to a prime with 3068389 decimal digits. - Serge Batalov, Jan 15 2018
a(19) <= 123447 corresponds to a prime with 5338805 decimal digits. - Serge Batalov, Jan 15 2018
a(20) <= 465859 corresponds to a prime with 11887192 decimal digits (not all lower candidates have been checked). This is the largest known non-Mersenne prime at the time of its discovery. - Serge Batalov, May 31 2023

C. Caldwell, Generalized unique primes
C. Caldwell, The Prime Pages, Phi_3(-465859^1048576)

Cf. A056993, A085398, A101406, A153436, A153438, A205506, A246120, A246121, A298206.

Table[SelectFirst[Range@ 200, PrimeQ[#^(2^n) (#^(2^n) - 1) + 1] &], {n, 0, 9}] (* Michael De Vlieger, Jan 15 2018 *)

a(n)=k=1;while(!ispseudoprime(k^(2^n)*(k^(2^n)-1)+1),k++);k
n=0;while(n<100,print1(a(n),", ");n++) \\ Derek Orr, Aug 14 2014

a(n) = A085398(3*2^(n+1)). - Jinyuan Wang, Jan 01 2023

a(16) from Serge Batalov, Dec 30 2014
a(17) from Serge Batalov, Feb 10 2015
a(18-19) from Serge Batalov, May 31 2023

A246120 Least k such that k^(3^n)*(k^(3^n) - 1) + 1 is prime.

Original entry on oeis.org

2, 6, 7, 93, 15, 372, 421, 759, 7426, 9087

Offset: 0

Serge Batalov, Aug 14 2014

Numbers of the form k^m*(k^m - 1) + 1 with m > 0, k > 1 may be primes only if m is 3-smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors > 3. This sequence is a subset of A205506 with only m=3^n, which is similar to A153438.

Search limits: a(10) > 35000, a(11) > 3500.

			When k = 7, k^18 - k^9 + 1 is prime. Since this isn't prime for k < 7, a(2) = 7.

Cf. A056993, A085398, A101406, A153436, A153438, A205506, A246119, A246121.

a246120[n_Integer] := Module[{k = 1},
  While[! PrimeQ[k^(3^n)*(k^(3^n) - 1) + 1], k++]; k]; a246120 /@ Range[0, 9] (* Michael De Vlieger, Aug 15 2014 *)

a(n)=k=1;while(!ispseudoprime(k^(3^n)*(k^(3^n)-1)+1),k++);k
n=0;while(n<100,print1(a(n),", ");n++) \\ Derek Orr, Aug 14 2014

a(n) = A085398(2*3^(n+1)). - Jinyuan Wang, Jan 01 2023

A246121 Least k such that k^(6^n)*(k^(6^n) - 1) + 1 is prime.

Original entry on oeis.org

2, 3, 88, 28, 688, 7003, 1925

Offset: 0

Serge Batalov, Aug 14 2014

Numbers of the form k^m*(k^m - 1) + 1 with m > 0, k > 1 may be primes only if m is 3-smooth, because these numbers are Phi(6,k^m) and cyclotomic factorizations apply to any prime divisors > 3. This sequence is a subset of A205506 with only m=6^n.

Numbers of this form are Generalized unique primes. a(6) generates a 306477-digit prime.

			When k = 88, k^72 - k^36 + 1 is prime. Since this isn't prime for k < 88, a(2) = 88.

C. Caldwell, Generalized unique primes

Cf. A056993, A085398, A101406, A153436, A153438, A205506, A246119, A246120.

a(n)=k=1; while(!ispseudoprime(k^(6^n)*(k^(6^n)-1)+1), k++); k
n=0; while(n<100, print1(a(n), ", "); n++)

a(n) = A085398(6^(n+1)). - Jinyuan Wang, Jan 01 2023

a(6) from Serge Batalov, Aug 15 2014

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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A153439 Numbers k such that k^6 + k^3 + 1 is prime.

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A153440 Numbers k such that k^9*(k^9+1)+1 is prime.

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Magma

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A205506 Least positive integer m > 1 such that 1 - m^k + m^(2*k) is prime, where k=A003586(n).

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A153436 a(n) is the least k such that k^n*(k^n+1)-1 is prime.

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A153441 Numbers k such that k^27*(k^27+1)+1 is prime.

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A153442 Numbers k such that k^81*(k^81+1)+1 is prime.

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Mathematica

A246119 a(n) is the least k such that k^(2^n)*(k^(2^n) - 1) + 1 is prime.

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