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%I #13 Jun 17 2021 04:44:43
%S 7,17,2,5,149,263,389,11,449,821,1091,881,1913,23,2729,29,2531,6599,
%T 2591,6971,3989,41,4583,5189,7019,7673,53,13679,7853,8699,12641,9521,
%U 13691,12143,19403,13109,22901,83,18251,89,20543,32183,23063,26693
%N Least initial value for a Euclid/Mullin sequence whose 3rd term (= least prime divisor of 1+2p) equals the n-th prime. prime(1)=2 is never a third term, so offset=2.
%C These primes are congruent to (prime(n)-1)/2 mod prime(n) if n > 4. Presumably all primes occur as 3rd term if initial prime is suitably chosen.
%F a(n) = Min[x; A094460(x) = prime(n)].
%e n=25: prime(25) = 97 and an Euclid-Mullin sequence started with a(25) = 5189 = 97*53 + 48 is {5189, 2, 97, 101, 3, 7, 167, 356568143863}.
%e All larger (prime) solutions with 97 as 3rd term have the form 97k + 48 form. However, not all primes of the form 97k + 48 result in Euclid-Mullin (EM) sequences with the property that the 3rd term is 97. For example, 727 = 7*97 + 48 is a prime providing an EM sequence as follows: {727, 2, 3, 4363, 19, 5, 1709, 11, 33988283132431, 7} with 3rd term = 3.
%e Analogous statements hold for other initial or 3rd primes.
%t a[x_]:=First[Flatten[FactorInteger[Apply[Times, Table[a[j], {j, 1, x-1}]]+1]]]; ta=Table[0, {20000}];a[1]=1;Do[{a[1]=Prime[j], el=3}; ta[[j]]=a[el], {j, 1, 20000}] Table[Prime[Min[Flatten[Position[ta, Prime[w]]]]], {w, 1, 100}]
%Y Cf. A000945, A051308-A051334, A094460.
%K nonn
%O 2,1
%A _Labos Elemer_, May 10 2004
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