A390117 Least integer m > 1 such that m*n + 1 is a prime.

2, 2, 2, 3, 2, 2, 4, 2, 2, 3, 2, 3, 4, 2, 2, 6, 6, 2, 10, 2, 2, 3, 2, 3, 4, 2, 4, 4, 2, 2, 10, 3, 2, 3, 2, 2, 4, 5, 2, 6, 2, 3, 4, 2, 4, 3, 6, 2, 4, 2, 2, 3, 2, 2, 6, 2, 4, 4, 12, 3, 6, 5, 2, 3, 2, 3, 4, 2, 2, 3, 8, 6, 4, 2, 2, 3, 6, 2, 4, 3, 2, 9, 2, 4, 12, 2, 4, 4, 2, 2, 6, 3, 4, 3, 2, 2, 4, 2, 2, 4

Offset

1,1

Comments

Clearly, a(n) >= A034693(n).

Conjecture: (i) a(n) <= 2*sqrt(n+6) for all n > 0.

(ii) For any n > 0, there is a positive integer m <= 2*sqrt(n+3) such that m*n - 1 is prime.

If p = m*n + 1 is prime with 1 < m <= n, then m*(n+1) = p + m - 1 and (m-1)*(n+1) = m*n + m - n - 1 < p, hence by Lucas' theorem we have C(m*(n+1)^2,m*(n+1)) = C(p*(n+1)+(m-1)(n+1),p+m-1) == C(n+1,1)*C((m-1)(n+1),m-1) (mod p) and thus p does not divide C(m*(n+1)^2,m*(n+1)).

Similarly, if p = m*n - 1 is an odd prime with m > 0, then by Lucas' theorem C(m*n^2,m*n) = C(p*n+n,p+1) is not divided by p.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, On divisibility of binomial coefficients, J. Austral. Math. Soc. 93 (2012), 189--201.

Yuqing He and Pingzhi Yuan, Results on some conjectures about binomial coefficients, Bull. Math. Soc. Sci. Math. Roumanie 67 (2024), 429--448.

Example

a(1) = 2 with 2*1 + 1 = 3 prime.
a(4) = 3 with 3*4 + 1 = 13 prime.

Mathematica

tab={}; Do[m=2; Label[bb]; If[PrimeQ[m*n+1], tab=Append[tab, m]; Goto[aa]]; m=m+1; Goto[bb]; Label[aa], {n, 1, 100}]; Print[tab]

Prog

(PARI) a(n) = my(m=2); while (!isprime(m*n + 1), m++); m; \\ Michel Marcus, Oct 26 2025

Crossrefs

Cf. A000040, A034693, A072668 (when a(n) = A034693(n)), A390107.

Sequence in context: A175778 A357039 A226182 * A099774 A305973 A290978

Adjacent sequences: A390114 A390115 A390116 * A390118 A390119 A390120

Keyword

nonn

Author

Zhi-Wei Sun, Oct 25 2025

Status

approved