A108795 Conjectured greatest number k such that C(2k,k) is not divisible by any odd prime to the n-th power.

1, 786, 538279, 1430148153

Offset: 1

Author

R. K. Guy and Robert G. Wilson v, Nov 29 2005

Keywords

• nonn

Comments

Checked by Jack Brennen to 5.93*10^10 and in fact every number beyond 14384056005 was divisible by at least two odd-prime-fourth-powers. C(2*14384056005,14384056005) seems to be the last such number which is only divisible by a single odd-prime-fourth-power, being divisible by 5^9 but by no other prime more than 3 times.

Examples

			a(1)=1 because for all k's>1 C(2k,k) is divisible by an odd prime.
a(2)=786 because it is the last entry in A059097, i.e., C(1572,786) has no prime factor squared.
		

References

• R. K. Guy, Unsolved Problems in Number Theory, C33.

Crossrefs

Cf. A059097.

Programs

Mathematica

expoPF[k_, n_] := Module[{s = 0, x = n}, While[x > 0, x = Floor[x/k]; s += x]; s]; goodQ[n_] := Module[{i = 2, p}, While[p = Prime[i]; p <= n && expoPF[p, 2n] < 3 + 2expoPF[p, n], i++ ]; p > n]; Do[ If[ goodQ[n], Print[n]], {n, 5500000}]

Extensions

a(4) from Jack Brennen, Nov 30 2005