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A084742
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Least k such that (n^k+1)/(n+1) is prime, or 0 if no such prime exists.
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21
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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
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OFFSET
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2,1
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COMMENTS
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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(311) = 2707, a(313) = 4451. - Eric Chen, Nov 20 2014
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
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LINKS
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Eric Weisstein's World of Mathematics, Repunit
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EXAMPLE
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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.
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Amarnath Murthy and Meenakshi Srikanth (menakan_s(AT)yahoo.com), Jun 15 2003
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EXTENSIONS
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STATUS
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approved
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