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A094464
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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.
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1
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7, 17, 2, 5, 149, 263, 389, 11, 449, 821, 1091, 881, 1913, 23, 2729, 29, 2531, 6599, 2591, 6971, 3989, 41, 4583, 5189, 7019, 7673, 53, 13679, 7853, 8699, 12641, 9521, 13691, 12143, 19403, 13109, 22901, 83, 18251, 89, 20543, 32183, 23063, 26693
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OFFSET
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2,1
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COMMENTS
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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.
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LINKS
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FORMULA
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a(n) = Min[x; A094460(x) = prime(n)].
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EXAMPLE
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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}.
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.
Analogous statements hold for other initial or 3rd primes.
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MATHEMATICA
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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}]
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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