A101406 -id:A101406 - cp's OEIS frontend

A153438 Least k > 1 such that k^(3^n)*(k^(3^n)+1) + 1 is prime.

2, 2, 2, 21, 209, 72, 260, 17, 3311, 4469, 94259, 55599

Offset: 0

Pierre CAMI, Dec 26 2008

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 is k^n*(k^n+1) + 1 always composite?
Phi[3^(n+1),k] = k^(3^n)*(k^(3^n)+1)+1. When m <> 3^n in k^m*(k^m+1)+1, Phi[3m,k] < k^m*(k^m+1)+1 and is a divisor of it. - Lei Zhou, Feb 09 2012
The prime number corresponding to the 10th term is a 587458-digit number. - Lei Zhou, Jul 04 2014
x^(2*k) + x^k + 1 = (x^(3*k) - 1)/(x^k - 1) is the product over n dividing 3k but not dividing k of cyclotomic polynomials Phi(n). If k is a power of 3, n = 3k is the only such divisor and we have a single irreducible cyclotomic polynomial Phi(3k). Otherwise we have the product of more than one polynomial, with integer values > 1 for integer x > 1, and thus always composite numbers. - Martin Becker, Jun 22 2021

Lei Zhou, Prime Database Entry, July 04, 2014.
R. Propper and S. Batalov, Prime Database Entry, Nov 17, 2023.

Cf. A056993, A085398, A101406, A153436.

Table[i = 1; m = 3^u; While[i++; cp = 1 + i^m + i^(2*m); ! PrimeQ[cp]]; i, {u, 1, 7}] (* Lei Zhou, Feb 01 2012 *)

a(n) = my(k=2); while (!isprime(k^(3^n)*(k^(3^n)+1) + 1), k++); k; \\ Michel Marcus, Jan 01 2023

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

3311 from Lei Zhou using OpenPFGW, Feb 01 2012
4469 from Lei Zhou using OpenPFGW, Feb 09 2012
New term, 94259, from Lei Zhou using OpenPFGW, Jul 04 2014
Name and Comment corrected by Robert Price, Nov 11 2018
a(0) inserted by Jinyuan Wang, Jan 01 2023
a(11) from Ryan Propper and Serge Batalov, Nov 17 2023

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

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

A101446 a(n)= least k such that n^k*(n^k-1)-1 is prime with n > 1.

Original entry on oeis.org

2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 10, 1, 5, 1, 1, 2, 2, 1, 1, 1, 2, 4, 1, 4, 1, 12, 1, 328, 1, 1, 6, 24, 77, 1, 8, 75, 1, 1, 2, 1, 2, 13, 1, 1, 1, 14, 1, 3, 1, 45, 3, 1, 1, 1, 1, 8, 625, 1, 1, 22, 3, 3, 1, 1, 1, 11, 1, 4, 1, 2, 33803, 5, 10, 64, 1, 3, 8, 2, 5, 2, 2

Offset: 2

Pierre CAMI, Jan 24 2005

nonn

a(73) > 2475. - Ray Chandler, Jan 27 2005

			2^2*(2^2-1)-1=11 prime so a(2)=2.

Michael S. Branicky, Table of n, a(n) for n = 2..147

Cf. A101406.

f[n_] := Block[{k = 1, c}, While[c = n^k; ! PrimeQ[c*(c - 1) - 1], k++ ]; Return[k]; ]; Table[f[n], {n, 2, 72}] (* Ray Chandler, Jan 27 2005 *)

a(n) = {my(k=1); while (!ispseudoprime(n^k*(n^k-1)-1), k++); k;} \\ Michel Marcus, Sep 16 2019

a(55)-a(72) from Ray Chandler, Jan 27 2005

a(73) and beyond from Michael S. Branicky, Jun 04 2025

cp's OEIS Frontend

A153438 Least k > 1 such that k^(3^n)*(k^(3^n)+1) + 1 is prime.

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

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

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

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A101446 a(n)= least k such that n^k*(n^k-1)-1 is prime with n > 1.

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