A094465 Least initial value for an Euclid/Mullin sequence whose 4th term is prime(n). prime(1)=2 is never a fourth term, so offset=2.
5, 19, 43, 31, 67, 541, 193, 157, 1213, 811, 487, 2371, 2, 1543, 733, 1319, 1291, 1753, 1621, 2713, 13, 1231, 2833, 2053, 1801, 3313, 5011, 821, 2467, 5101, 3253, 8573, 3637, 1553, 15427, 5521, 3191, 9173, 7237, 10531, 11071, 6271, 9103, 15727, 7993
Offset: 2
Keywords
Examples
n=14: prime(14) = 43 and an Euclid-Mullin sequence started with a(14) = 2 = prime(1) is {2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, ...} is A000945, the prototype EM-sequence.
n=7: a(7) = prime(100) = 541, with EM sequence as follows: {541, 2, 3, 17, 139, 7, 1871, 100457892907, 19, 11047, ...}, where the 4th term equals prime(n) = prime(7) = 17.
It is characteristic but not so simple congruence relations holds of term(1) mod term(4) form for various first or 4th primes, not necessarily smallest ones. See comment at A094464.
Programs
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Mathematica
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=4}; ta[[j]]=a[el], {j, 1, 20000}] Table[Prime[Min[Flatten[Position[ta, Prime[w]]]]], {w, 1, 100}]
Formula
a(n) = Min_{k} A051614(k) = prime(n).