A242173 - OEIS

%I #28 Nov 03 2019 19:46:37

%S 3,13,7,107,11,89,31,265729,19,9887,23,113,79,373,53,3089,151,127,719,

%T 193,43,482673878761,47,61403,109,37889,1223,3251609,59,181,

%U 22504880485262968151,3598831,67,69593,179,13828116559,4247285503,1579,19095283759,619

%N 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.

%C Conjecture:

%C (i) a(n) > 1 for all n > 0.

%C (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.

%H Zhi-Wei Sun, <a href="/A242173/b242173.txt">Table of n, a(n) for n = 1..119</a>

%e a(3) = 7 since D(3) = 3^2*7 with 7 dividing none of D(1) = 3 and D(2) = 13.

%t d[n_]:=Sum[Binomial[n+k,k]*Binomial[n,k],{k,0,n}]

%t f[n_]:=FactorInteger[d[n]]

%t p[n_]:=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]

%t 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}]

%Y Cf. A000040, A001850, A005259, A242169, A242170, A242171.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, May 06 2014