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
Zhi-Wei Sun, Table of n, a(n) for n = 1..119
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