A080170 Numbers k such that gcd(C(2*k,k), C(3*k,k), C(4*k,k), ..., C((k+1)*k,k) ) = 1.

29, 59, 69, 83, 89, 104, 119, 125, 131, 139, 153, 164, 167, 179, 181, 194, 197, 209, 219, 230, 233, 251, 259, 263, 272, 279, 285, 305, 307, 311, 314, 329, 335, 339, 356, 359, 363, 373, 379, 384, 389, 395, 398, 407, 417, 419, 428, 439, 441, 454, 455, 461

Offset

1,1

Comments

a(n) = A051283(n) - 1 (conjectured). - Ralf Stephan, Feb 20 2004

The main result of Guo et al. is proof of the above conjecture. A Lean formalization is given in the paper. - Michael De Vlieger, Jun 29 2026

Links

Table of n, a(n) for n=1..52.

Dakai Guo, Ruichen Qiu, Yichuan Cao, Ruyong Feng, and Xiao-Shan Gao, A Greatest Common Divisor Criterion of Certain Binomial Coefficients, arXiv:2606.22997 [math.NT], 2026.

Mathematica

Select[Range[500], GCD@@Table[Binomial[k*#, # ], {k, 2, #+1}]==1&]

Prog

(Haskell)
a080170 n = a080170_list !! (n-1)
a080170_list = filter f [1..] where
   f x = foldl1 gcd (map (flip a007318' x) [2*x, 3*x .. x*(x+1)]) == 1
-- Reinhard Zumkeller, May 30 2013
(PARI) isok(k) = gcd(vector(k, i, binomial(k+i*k, k))) == 1; \\ Jinyuan Wang, Feb 28 2020

Crossrefs

Cf. A007318, A051283.

Sequence in context: A004942 A033904 A184073 * A078848 A055784 A083821

Adjacent sequences: A080167 A080168 A080169 * A080171 A080172 A080173

Keyword

nonn

Author

Benoit Cloitre, Jan 31 2003

Status

approved