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A035096
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a(n) is the smallest k such that prime(n)*k+1 is prime.
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14
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1, 2, 2, 4, 2, 4, 6, 10, 2, 2, 10, 4, 2, 4, 6, 2, 12, 6, 4, 8, 4, 4, 2, 2, 4, 6, 6, 6, 10, 2, 4, 2, 6, 4, 8, 6, 10, 4, 14, 2, 2, 6, 2, 4, 18, 4, 10, 12, 24, 12, 2, 2, 6, 2, 6, 6, 8, 6, 4, 2, 6, 2, 4, 6, 6, 26, 6, 10, 6, 10, 14, 2, 6, 4, 12, 12, 24, 6, 8, 4, 2, 10, 2, 4, 10, 2, 8, 30
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OFFSET
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1,2
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COMMENTS
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These arithmetic progressions have prime differences. Note that both the terms of generated by this k values and the differences are primes as well.
This is one possible generalization of "the least prime problem in special arithmetic progressions" when n in the nk+1 form is replaced by n-th prime number.
Note that Dirichlet's theorem on primes in arithmetic progressions implies that a(n) always exists. - Max Alekseyev, Jul 11 2008
If a(n)=2, prime(n) is a Sophie Germain prime (A005384). Among the first 10^6 terms, the largest is a(330408) = 234. - Zak Seidov, Jan 28 2012
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LINKS
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FORMULA
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EXAMPLE
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a(15)=6 because the 15th prime is 47, and the smallest k such that 47k+1 is prime is k=6, for which 47k+1=283.
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MATHEMATICA
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Reap[Sow[1]; Do[p = Prime[n]; k = 2; While[! PrimeQ[k*p + 1], k = k + 2]; Sow[k], {n, 2, 10^4}]][[2, 1]] (* Zak Seidov, Jan 28 2012 *)
f[n_] := Block[{p = Prime@ n}, q = 1 + 2p; While[ !PrimeQ@ q, q += 2p]; (q - 1)/p]; f[1] = 1; Array[f, 88] (* Robert G. Wilson v, Dec 27 2014 *)
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PROG
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(Magma)
S:=[];
k:=1;
for n in [1..90] do
while not IsPrime(k*NthPrime(n)+1) do
k:=k+1;
end while;
Append(~S, k);
k:=1;
end for;
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CROSSREFS
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Smallest k such that k*n+1 is prime is A034693.
Sophie Germain primes are in A005384.
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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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