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A242173
Least prime divisor of the n-th central Delannoy number D(n) which does not divide any D(k) with k < n, or 1 if such a primitive prime divisor of D(n) does not exist.
11
3, 13, 7, 107, 11, 89, 31, 265729, 19, 9887, 23, 113, 79, 373, 53, 3089, 151, 127, 719, 193, 43, 482673878761, 47, 61403, 109, 37889, 1223, 3251609, 59, 181, 22504880485262968151, 3598831, 67, 69593, 179, 13828116559, 4247285503, 1579, 19095283759, 619
OFFSET
1,1
COMMENTS
Conjecture:
(i) a(n) > 1 for all n > 0.
(ii) For any integer n > 0, the n-th Apéry number A(n) = Sum_{k=0..n} (binomial(n,k)*binomial(n+k,k))^2 has a prime divisor which does not divide any A(k) with k < n.
LINKS
EXAMPLE
a(3) = 7 since D(3) = 3^2*7 with 7 dividing none of D(1) = 3 and D(2) = 13.
MATHEMATICA
d[n_]:=Sum[Binomial[n+k, k]*Binomial[n, k], {k, 0, n}]
f[n_]:=FactorInteger[d[n]]
p[n_]:=Table[Part[Part[f[n], k], 1], {k, 1, Length[f[n]]}]
Do[If[d[n]<2, Goto[cc]]; Do[Do[If[Mod[d[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 06 2014
STATUS
approved