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A242170
Least prime divisor of T(n) which does not divide any T(k) with k < n, or 1 if such a primitive prime divisor of T(n) does not exist, where T(n) is the n-th central trinomial coefficient given by A002426.
11
1, 3, 7, 19, 17, 47, 131, 41, 43, 1279, 503, 113, 2917, 569, 198623, 14083, 26693, 201611, 42998951, 41931041, 52635749, 1296973, 169097, 1451, 1304394227, 107, 233, 173, 2062225210273, 719, 191, 31551555041, 6301, 563, 3769, 967, 9539, 5073466546857451, 4542977, 9739
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
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