A239735 Least number k such that n*k^n +/- 1 are twin primes, or a(n) = 0 if no such number exists.

4, 3, 4, 1, 570, 1, 1464, 54, 60, 14025, 1932, 1, 7194, 15, 3612, 0, 4746, 1, 540, 150, 7060, 138, 80094, 6160, 33480, 93135, 0, 366618, 26058, 1, 90510, 16836, 9824, 418875, 57246, 0, 182394, 64077, 14178, 943410, 36078, 1, 314520, 15870, 194942, 15044700, 241944, 3871, 308730

Offset

1,1

Comments

a(n) = 1 iff n is in A014574.

If a(n) = 0, then n is in A097764.

If a(n) > 1 then A367566(n) divides a(n). - Jon E. Schoenfield, Nov 23 2023

Links

Jon E. Schoenfield, Table of n, a(n) for n = 1..200 (first 77 terms from Michel Marcus)

Example

1*1^1+1 (2) and 1*1^1-1 (0) are not both prime. 1*2^1+1 (3) and 1*2^1-1 (1) are not both prime. 1*3^1+1 (4) and 1*3^1-1 (2) are not both prime. 1*4^1+1 (5) and 1*4^1-1 (3) are both prime. So, a(1) = 4.

Mathematica

zeroQ[n_] := Module[{f = FactorInteger[n]}, pow = GCD @@ f[[;; , 2]]; n > 4 && AnyTrue[Divisors[pow], # > 1 && Divisible[n, #] &]];
a[n_, kmax_] := Module[{k = 1}, If[zeroQ[n], 0, While[k <= kmax && ! And @@ PrimeQ[n*k^n + {-1, 1}], k++]; If[k < kmax, k, -1]]]; Table[a[n, 10^6], {n, 1, 25}] (* Amiram Eldar, Nov 18 2023, returns -1 if the search limit should exceed kmax *)

Prog

(PARI) bot(n) = for(k=1, 10^5, if(ispseudoprime(n*k^n-1), if(ispseudoprime(n*k^n+1), return(k))));
n=1; while(n<100, print1(bot(n), ", "); n+=1)
(PARI) a(n) = if ((n==16) || (n==27) || (n==36) || (n==64) /* || (n== ... */, return(0)); my(k=1); while (!(ispseudoprime(n*k^n-1) && ispseudoprime(n*k^n+1)), k++); k; \\ Michel Marcus, Nov 18 2023

Crossrefs

Cf. A097764, A014574, A239938, A239666, A367566.

Sequence in context: A284517 A286953 A170987 * A196826 A215597 A266110

Adjacent sequences: A239732 A239733 A239734 * A239736 A239737 A239738

Keyword

nonn

Author

Derek Orr, Mar 30 2014

Extensions

a(46) from Giovanni Resta, Mar 31 2014

Status

approved