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A242169 Least prime divisor of Fr(n) which does not divide any Fr(k) with k < n, or 1 if such a primitive prime divisor of Fr(n) does not exist, where Fr(n) denotes the n-th Franel number given by A000172. 11

%I

%S 2,5,7,173,563,13,41,369581,937,61,23,29,2141,12148537,31,157,59,37,

%T 506251,151,3019,769,47,6730949,79,53,3853,661,138961158000728258971,

%U 1361,421,96920594213,51378681049,457,71

%N Least prime divisor of Fr(n) which does not divide any Fr(k) with k < n, or 1 if such a primitive prime divisor of Fr(n) does not exist, where Fr(n) denotes the n-th Franel number given by A000172.

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

%H Zhi-Wei Sun, <a href="/A242169/b242169.txt">Table of n, a(n) for n = 1..105</a>

%e a(7) = 41 since Fr(7) = 2^9*5*41 with the prime factor 41 dividing none of Fr(1), ..., Fr(6) but 2 divides Fr(1) = 2 and 5 divides Fr(2) = 10.

%t Fr[n_]:=Sum[Binomial[n,k]^3,{k,0,n}]

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

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

%t Do[Do[Do[If[Mod[Fr[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]]}];Print[n," ",1];Label[bb];Continue,{n,1,35}]

%Y Cf. A000040, A000172, A242170, A242171, A242173.

%K nonn

%O 1,1

%A _Zhi-Wei Sun_, May 05 2014

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Last modified August 2 05:02 EDT 2021. Contains 346409 sequences. (Running on oeis4.)