A110496 Least k such that prime(n)^3 divides binomial(2k,k).

7, 14, 63, 172, 666, 1099, 2457, 3430, 6084, 12195, 14896, 25327, 34461, 39754, 51912, 74439, 102690, 113491, 150382, 178956, 194509, 246520, 285894, 352485, 456337, 515151, 546364, 612522, 647515, 721449, 1024192, 1124046, 1285677

Offset

1,1

Comments

For prime p > (2n)^(1/3), p^3 does not divide binomial(2n,n).

Links

Chai Wah Wu, Table of n, a(n) for n = 1..100

Formula

a(n) = (prime(n)^3 + 1)/2 for n>1.

Product_{n>=1} (1 - 1/a(n)) = (54/49)*zeta(6)/zeta(3)^2. - Amiram Eldar, Jun 08 2022

Mathematica

t3=Table[f=FactorInteger[Binomial[2n, n]]; s=Select[f, #[[2]]>2&]; If[s=={}, 0, s[[ -1, 1]]], {n, 15000}]; Table[p=Prime[i]; First[Flatten[Position[t3, p]]], {i, PrimePi[Max[t3]]}]
lst={7}; Do[AppendTo[lst, (DivisorSigma[3, Prime[n]])/2], {n, 2, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)

Crossrefs

Cf. A000984, A030078.

Cf. A110495 (binomial(2k, k) is cubefree).

Sequence in context: A241201 A295388 A020700 * A117867 A291008 A196254

Adjacent sequences: A110493 A110494 A110495 * A110497 A110498 A110499

Keyword

nonn

Author

T. D. Noe, Jul 22 2005

Status

approved