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A242174 Least prime divisor of A005260(n) which does not divide any previous term A005260(k) with k < n, or 1 if such a primitive prime divisor of A005260(n) does not exist.

%I A242174 #20 May 07 2014 05:43:07
%S A242174 2,3,41,5,7,349,61,75617,31,13,499,643897693,17,19,1729774061,101,
%T A242174 2859112064587,138407,83,167,59,29,653,257,997540809461453561581,347,
%U A242174 13679,37,160449179727717672892660463,211,151,43,97,73,47
%N A242174 Least prime divisor of A005260(n) which does not divide any previous term A005260(k) with k < n, or 1 if such a primitive prime divisor of A005260(n) does not exist.
%C A242174 Conjecture: a(n) is prime for any n > 0. In general, for any r > 2, if n is large enough then f_r(n) = sum_{k=0..n}C(n,k)^r has a prime divisor which does not divide any previous terms f_r(k) with k < n.
%H A242174 Zhi-Wei Sun, <a href="/A242174/b242174.txt">Table of n, a(n) for n = 1..82</a>
%e A242174 a(3) = 41 since A005260(3) = 2^2*41 with 41 dividing none of A005260(1) = 2 and A005260(2) = 2*3^2.
%t A242174 u[n_]:=Sum[Binomial[n,k]^4,{k,0,n}]
%t A242174 f[n_]:=FactorInteger[u[n]]
%t A242174 p[n_]:=Table[Part[Part[f[n],k],1],{k,1,Length[f[n]]}]
%t A242174 Do[If[u[n]<2,Goto[cc]];Do[Do[If[Mod[u[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,1,35}]
%Y A242174 Cf. A000040, A005260, A242169, A242170, A242171, A242173, A242193, A242194, A242195, A242207.
%K A242174 nonn
%O A242174 1,1
%A A242174 _Zhi-Wei Sun_, May 07 2014