A390107 Least positive integer m such that m*(n-1) + 1 is relatively prime to C(m*n^2, m*n), where C(k, j) denotes the binomial coefficient k!/(j!*(k-j)!).

1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 3, 6, 2, 2, 2, 2, 3, 2, 2, 4, 2, 2, 2, 2, 2, 10, 3, 2, 3, 2, 2, 4, 5, 2, 2, 2, 3, 4, 2, 4, 3, 6, 2, 4, 2, 2, 3, 2, 2, 2, 2, 4, 4, 2, 2, 6, 2, 2, 3, 2, 3, 4, 2, 2, 3, 8, 4, 4, 2, 2, 3, 2, 2, 4, 3, 2, 9, 2, 2, 2, 2, 4, 4, 2, 2, 6, 3, 4, 3, 2, 2, 4, 2, 2

Offset

1,2

Comments

Conjecture: a(n) <= sqrt(3*n + 4) for all n > 0.

This has been verified for n <= 30000.

In the linked 2012 paper, the author conjectured that for any integer k > 2 there is a positive integer n such that C(k*n, n) does not divide C(k*n, n - 1)*C(k^2*n, k*n) (i.e., (k - 1)*n + 1 does not divide C(k^2*n, k*n)).

See also A390117 for a related conjecture.

Links

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

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

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

Example

a(3) = 2 with 2*(3 - 1) + 1 = 5 relatively prime to C(2*3^2,2*3) = 18564.
a(5) = 2 with 2*(5 - 1) + 1 = 9 relatively prime to C(2*5^2,2*5) = 10272278170.

Mathematica

tab={}; Do[m=1; Label[bb]; If[GCD[Binomial[m*n^2, m*n], m*(n-1)+1]==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=1); while (gcd(m*(n-1) + 1, binomial(m*n^2, m*n)) != 1, m++); m; \\ Michel Marcus, Oct 25 2025

Crossrefs

Cf. A000290, A390117.

Sequence in context: A358473 A393840 A369978 * A083261 A003648 A003642

Adjacent sequences: A390104 A390105 A390106 * A390108 A390109 A390110

Keyword

nonn

Author

Zhi-Wei Sun, Oct 25 2025

Status

approved