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.

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

Author

Zhi-Wei Sun, May 06 2014

Keywords

• nonn

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.

Examples

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

Links

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

Crossrefs

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

Programs

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}]