%I #10 Nov 05 2013 20:57:30
%S 1,786,538279,1430148153
%N Conjectured greatest number k such that C(2k,k) is not divisible by any odd prime to the n-th power.
%C 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.
%D R. K. Guy, Unsolved Problems in Number Theory, C33.
%e a(1)=1 because for all k's>1 C(2k,k) is divisible by an odd prime.
%e a(2)=786 because it is the last entry in A059097, i.e., C(1572,786) has no prime factor squared.
%t 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}]
%Y Cf. A059097.
%K nonn
%O 1,2
%A _R. K. Guy_ and _Robert G. Wilson v_, Nov 29 2005
%E a(4) from _Jack Brennen_, Nov 30 2005
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