A392019 If binomial(n,k) = u*v where the only primes dividing u are <= k and the only primes dividing v are > k, then a(n) is the least k such that u > n^2, or 0 if such a k doesn't exist.

0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 11, 0, 13, 0, 0, 8, 13, 0, 17, 12, 13, 13, 5, 17, 7, 19, 9, 12, 11, 11, 7, 11, 17, 15, 14, 17, 17, 19, 13, 13, 19, 23, 17, 17, 7, 7, 8, 15, 7, 8, 8, 9, 11, 18, 12, 14, 19, 16, 11, 12, 12, 13, 15, 19, 17, 20, 13, 19

Offset

1,10

Links

Table of n, a(n) for n=1..73.

Thomas Bloom, Problem #684, Erdős Problems.

Paul Erdős, Some unconventional problems in number theory, Acta Math. Acad. Sci. Hungar. (1979), 71-80.

Example

a(10)=7 because binomial(10,7) = 120 and 120 = 2^3*3*5 > 100 = 10^2.
a(21)=8 because binomial(21,8) = 203490 = 630*323 and 630 = 2*3^2*5*7 > 441 = 21^2.

Mathematica

u[n_, k_]:=Times@@Power@@@Select[FactorInteger[Binomial[n, k]], #[[1]]<=k&];
a[n_]:=Append[Select[u[n, #]&/@Range[n], #>n^2&->"Index", 1], 0][[1]];

Prog

(PARI)
a(n)={my(uv, f, u); for(k=0, n, uv=binomial(n, k);
  f=factor(uv); u=1; for(i=1, matsize(f)[1], if(f[i, 1]<=k, u*=f[i, 1]^f[i, 2]; ); );
  if(u>n^2, return(k); ); ); return(0); };

Crossrefs

Cf. A007318.

Sequence in context: A106226 A229658 A306755 * A005070 A135435 A339442

Adjacent sequences: A392016 A392017 A392018 * A392020 A392021 A392022

Keyword

nonn

Author

Elijah Beregovsky, Dec 27 2025

Extensions

a(7) corrected by Tomás Oliveira e Silva, Apr 29 2026

Further terms corrected by Sean A. Irvine, May 22 2026

Status

approved