A087985 a(n) = prime(x) is the smallest prime such that 1+n*prime(x) is divisible by prime(x+1), or 0 if no such prime exists.

2, 0, 3, 2, 23, 19, 2, 3, 47, 2, 5, 43, 2, 7, 83, 2, 0, 3, 2, 11, 79, 2, 3, 0, 2, 0, 103, 2, 17, 13, 2, 5, 3, 2, 0, 7, 2, 3, 5, 2, 0, 163, 2, 257, 263, 2, 7, 3, 2, 0, 0, 2, 3, 0, 2, 61, 223, 2, 11, 5, 2, 47, 3, 2, 41, 73, 2, 3, 7, 2, 317, 11, 2, 5, 19, 2, 0, 3, 2, 7, 5, 2, 3, 829, 2, 17, 0, 2, 67, 0

Offset

1,1

Comments

Two unsolved problems:

1. Given arbitrary n, does there exist an m such that 1 + n*prime(x) = m*prime(x+1) is solvable? For several n (e.g., n = 17, 24, 26, 35, 41, 50, 51, 54, 87) a search up to x=10000000 did not provide a prime(x) solution.

2. If a solution exists, then the number of solutions is believed to be finite. E.g., at n=2122, 8 solutions were found after an extensive search {prime(x)} = {2, 19, 23, 37, 89, 433, 4241, 7621}.

Links

Table of n, a(n) for n=1..90.

Formula

a(n) = Min{prime(x); (1 + n*prime(x)) mod prime(x+1) = 0}; a(n) = p; 1 + n*p = m*q where {p, q} are smallest consecutive primes satisfying relation belonging to n.

Example

n=1: 1 + 1*p = q is satisfied by the consecutive prime pair {2,3}; a(1)=2.
n=2: 1 + 2*p = m*q, no solution.
n=6: 1 + 6*p = m*q is satisfied first by {p,q} = {19,23} because 6*19 + 1 = 115 = 5*23.
It is provable that there are arbitrarily large n to which (a finite number of) solutions exist.

Mathematica

{k=0, nu=0; sq={}}; Table[Print[{n-1, Min[Prime[sq]]}]; nu=0; sq={}; Do[s=Mod[n*Prime[x]+1, Prime[x+1]]; If[Equal[s, 0], nu=nu+1; sq=Append[sq, n]], {x, 1, 10000000}], {n, 1, 257}]

Prog

(PARI) a(n) = local(i, c, p, q); i = 1; c = 0; q = 2; while (1, p = q; q = prime(i + 1); if (!((1 + n*p)%q), return(p)); if ((i + n*p)/q > n - 1/10, c++; if (c == 8, return(0)), c = 0); i++); \\ David Wasserman, Jun 17 2005

Crossrefs

Cf. A087986.

Sequence in context: A079451 A219187 A049799 * A200927 A372820 A372819

Adjacent sequences: A087982 A087983 A087984 * A087986 A087987 A087988

Keyword

nonn

Author

Labos Elemer, Oct 06 2003

Extensions

More terms from David Wasserman, Jun 17 2005

Status

approved