

A059411


a(1) = 2, a(n) = k*a(n1) + 1, where a(n) is the smallest prime of the form k*a(n1) + 1 and k > 1.


7



2, 5, 11, 23, 47, 283, 1699, 20389, 244669, 7340071, 205521989, 411043979, 4932527749, 295951664941, 4735226639057, 227290878674737, 12273707448435799, 883706936287377529, 24743794216046570813
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OFFSET

1,1


COMMENTS

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 "almostquotientsequence".
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
The existence of a prime of the form k*p+1 follows from Dirichlet's theorem (1837).  T. D. Noe, Mar 14 2009


REFERENCES

Amarnath Murthy, On the divisors of Smarandache Unary Sequence. Smarandache Notions Journal, Vol. 11, No. 123, Spring 2000.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..100


FORMULA

a(n+1) = a(n)*q(n) + 1, q(n) = Min{qqa(n)+1 is prime}.


EXAMPLE

a(5) = 47 and a(6) = 283 = 6*47 +1 is the smallest such prime.
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 "quotientsequence" 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.


MAPLE

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[i1]+1; end do:q := seq(a[i], i=0..39);


MATHEMATICA

nxt[n_]:=Module[{k=2}, While[!PrimeQ[k*n+1], k++]; k*n+1]; NestList[nxt, 2, 20] (* Harvey P. Dale, Dec 26 2014 *)


CROSSREFS

Cf. A061092.
Sequence in context: A055011 A007505 A246492 * A126017 A292937 A034468
Adjacent sequences: A059408 A059409 A059410 * A059412 A059413 A059414


KEYWORD

nonn


AUTHOR

Labos Elemer, Jan 30 2001


STATUS

approved



