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

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

Author

Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 15 2003

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• 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.

Crossrefs

Cf. A084741, A065507, A084740, A128164, A126659, A103795.

Programs

PARI

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

Extensions

More terms from T. D. Noe, Jan 22 2004