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%I #13 Jul 07 2018 19:19:24
%S 2,5,11,23,47,283,1699,20389,244669,7340071,205521989,411043979,
%T 4932527749,295951664941,4735226639057,227290878674737,
%U 12273707448435799,883706936287377529,24743794216046570813
%N a(1) = 2, a(n) = k*a(n-1) + 1, where a(n) is the smallest prime of the form k*a(n-1) + 1 and k > 1.
%C A sequence of primes generated recursively as follows: a(n+1) = q(n)*a(n)+1, where q=q(n) is the smallest (even) number such that a(n+1) = q*a(n)+1 is prime and the initial value a(1)=2. q(n) = (a(n+1) - 1)/a(n) is the satellite "almost-quotient-sequence".
%C It has been established in the Murthy reference that for every prime p there exists at least one prime of the form k*p +1. Hence the sequence is infinite. - _Amarnath Murthy_, Mar 02 2002
%C The existence of a prime of the form k*p+1 follows from Dirichlet's theorem (1837). - _T. D. Noe_, Mar 14 2009
%D Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.
%H T. D. Noe, <a href="/A059411/b059411.txt">Table of n, a(n) for n = 1..100</a>
%F a(n+1) = a(n)*q(n) + 1, q(n) = Min{q|qa(n)+1 is prime}.
%e a(5) = 47 and a(6) = 283 = 6*47 +1 is the smallest such prime.
%e The initial values are safe primes: (2), 5, 11, 23, 47, ... To obtain qa(i)+1 primes q > 2 multiplier arises and such a q always exists in arithmetic progression of difference a(i). E.g., {1699*2k+1} gives the first prime when 2k=12. So a(7)=1699 is followed by 1699*12+1 = 20389 = a(8). The emergent "quotient-sequence" is {2, 2, 2, 2, 6, 6, 12, 12, 30, 28, 2, 12, 60, 16, 48, 54, 72, 28, 180, 102, 4, 12, 106, 50, 18}. A059411 is an infinite sequence of primes increasing at least with exponential speed.
%p i := 0:a[0] := 2:while(i<40) do k := 2:while(not isprime(k*a[i]+1)) do k := k+1; end do; i := i+1; a[i] := k*a[i-1]+1; end do:q := seq(a[i],i=0..39);
%t nxt[n_]:=Module[{k=2},While[!PrimeQ[k*n+1],k++];k*n+1]; NestList[nxt,2,20] (* _Harvey P. Dale_, Dec 26 2014 *)
%Y Cf. A061092.
%K nonn
%O 1,1
%A _Labos Elemer_, Jan 30 2001
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