

A035096


a(n) is the smallest k such that prime(n)*k+1 is prime.


11



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

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 nth 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


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Math, Dirichlet's theorem
Index entries for sequences related to primes in arithmetic progressions


FORMULA

a(n) = (A035095(n)1)/A000040(n).  Zak Seidov, Dec 27 2013


EXAMPLE

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.


MATHEMATICA

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 *)


PROG

(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;
S; // Bruno Berselli, Apr 18 2013


CROSSREFS

Smallest k such that k*n+1 is prime is A034693.
Sophie Germain primes are in A005384.
Cf. A000040, A035095.  Zak Seidov, Dec 27 2013
Sequence in context: A233971 A170905 A233761 * A066675 A219433 A274879
Adjacent sequences: A035093 A035094 A035095 * A035097 A035098 A035099


KEYWORD

nonn


AUTHOR

Labos Elemer


STATUS

approved



