STATUS
proposed
approved
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Revision History for A059411
(Bold, blue-underlined text is an addition; faded, red-underlined text is aShowing entries 1-10 | older changes
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.
(history; published version)
(history; published version)
STATUS
editing
proposed
NAME
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.
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 "almost-quotient-sequence".
The existence of a prime of the form k*p+1 follows from Dirichlet's theorem (1837). - _T. D. Noe, _, Mar 14 2009
LINKS
T. D. Noe, <a href="/A059411/b059411.txt">Table of n, a(n) for n = 1..100</a>
FORMULA
a(n+1) = a(n)*q(n) + 1, q(n) = Min{q|qa(n)+1 is prime}.
EXAMPLE
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.
STATUS
approved
editing
STATUS
editing
approved
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 *)
STATUS
approved
editing
COMMENTS
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 (amarnath_murthy(AT)yahoo.com), _, Mar 02 2002
AUTHOR
_Labos E. (labos(AT)ana.sote.hu), Elemer_, Jan 30 2001
LINKS
T. D. Noe, <a href="/A059411/b059411.txt">Table of n, a(n) for n=1..100</a>
KEYWORD
nonn,new
nonn
COMMENTS
The existence of a prime of the form k*p+1 follows from Dirichlet's theorem (1837). - T. D. Noe, Mar 14 2009
LINKS
T. D. Noe, <a href="b059411.txt">Table of n, a(n) for n=1..100</a>
KEYWORD
nonn,new
nonn
KEYWORD
nonn,new
nonn
AUTHOR
Labos E. (labos(AT)ana1ana.sote.hu), Jan 30 2001