OFFSET
1,2
COMMENTS
Conjecture: (i) a(n) > 1 for all n > 1.
(ii) For any integer n > 3, the n-th Motzkin number M(n) given by A001006 has a prime divisor which does not divide any M(k) with k < n.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..168
EXAMPLE
a(11) = 503 since T(11) = 3*17*503 with the prime divisor 503 dividing none of T(1),...,T(10), but 3 divides T(2) = 3 and 17 divides T(5) = 51.
MATHEMATICA
T[n_]:=Sum[Binomial[n, 2k]*Binomial[2k, k], {k, 0, n/2}]
f[n_]:=FactorInteger[T[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[T[n]<2, Goto[cc]]; Do[Do[If[Mod[T[i], Part[p[n], k]]==0, Goto[aa]], {i, 1, n-1}];
Print[n, " ", Part[p[n], k]]; Goto[bb]; Label[aa]; Continue, {k, 1, Length[p[n]]}];
Label[cc]; Print[n, " ", 1]; Label[bb]; Continue, {n, 1, 40}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, May 05 2014
STATUS
approved