Claims

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2024_07_29_stephan: The OEIS entry A375077, authored by Ralf Stephan, gives the least n with n(n-1)...(n-k) dividing the central binomial coefficient for k = 1, 2, 3, 4, namely 2, 2480, 8178 and 45153, so those four instances have the answer yes.

2024_08_01_wu: Chai Wah Wu's term a(5) = 3648841 of the OEIS entry A375077 is the least n with n(n-1)...(n-5) dividing the central binomial coefficient, so the instance k = 5 has the answer yes.

2025_02_25_alekseyev: Max Alekseyev's terms a(6) = 7979090 and a(7) = 101130029 of the OEIS entry A375077 are the least n with n(n-1)...(n-k) dividing the central binomial coefficient for k = 6 and 7, so those two instances have the answer yes.

2026_03_18_kesarwani: Sharvil Kesarwani's terms a(8) to a(11) and a(15) of the OEIS entry A375077 are the least n with n(n-1)...(n-k) dividing the central binomial coefficient for those k, so those instances have the answer yes.

2026_03_23_dehorty: Justin Dehorty's terms a(12), a(13) and a(16) of the OEIS entry A375077 are the least n with n(n-1)...(n-k) dividing the central binomial coefficient for those k, so those instances have the answer yes.

2026_04_03_dehorty_kesarwani: The term a(14) = 359503904702190 of the OEIS entry A375077, credited to Justin Dehorty and Sharvil Kesarwani, is the least n with n(n-1)...(n-14) dividing the central binomial coefficient, so k = 14 has the answer yes.

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