A103689 a(n) is the least k such that either k*n - 1 or k*n + 1 (or both) is prime.

1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 4, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 6, 1, 6, 1, 2, 2, 2, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 2, 4, 1, 2, 1, 2, 1, 2, 1, 6, 2, 2, 1, 2, 1, 2, 3, 4, 3, 2, 1, 2, 1, 2, 1, 6, 1, 6, 1, 2

Offset: 1

Author

Pierre CAMI, Feb 12 2005

Keywords

• easy
• nonn

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002110, A034693, A053989, A103688, A200996, A348347.

Programs

Haskell

a103689 n = min (a053989 n) (a034693 n)
-- Reinhard Zumkeller, Feb 14 2013

Mathematica

f[n_] := Block[{k = 1}, While[ ! PrimeQ[k*n - 1] && ! PrimeQ[k*n + 1], k++ ]; k]; Table[ f[n], {n, 105}] (* Robert G. Wilson v, Feb 12 2005 *)
lk[n_]:=Module[{k=1},While[NoneTrue[k*n+{1,-1},PrimeQ],k++];k]; Array[ lk,120] (* The program uses the NoneTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 01 2016 *)

PARI

a(n) = my(k=1); while (!isprime(k*n+1) && !isprime(k*n-1), k++); k; \\ Michel Marcus, Oct 18 2021

Formula

a(n) <= A200996(n). - Reinhard Zumkeller, Feb 14 2013

a(n) = min {A053989(n), A034693(n)}. - Reinhard Zumkeller, Feb 14 2013

a(A002110(n)/3+3) >= ceiling((prime(n+1)-1)/3) for n >= 2. Equality holds for n = 2, 4, 6, 8, 10, 12, 22, 25, 31, 116, 155, 156, 197, ... . - Pontus von Brömssen, Oct 16 2021

a(A002110(n)/3-3) >= ceiling((prime(n+1)-1)/3) for n >= 3. Equality holds for n = 3, 4, 5, 6, 7, 9, 39, 51, 59, 65, 98, 311, ... . - Pontus von Brömssen, Oct 19 2021

Extensions

Edited, corrected and extended by Robert G. Wilson v, Feb 19 2005