OFFSET
1,1
COMMENTS
The terms up to 1103 required examining numbers produced by Rowland's recurrence up to n = 10^8. - T. D. Noe, Apr 11 2013
Exactly 177789368686545736460055960459780707068552048703463291 iterations to find the first 1000 terms of this sequence. - T. D. Noe, Apr 13 2013
The first 10^100 terms of Rowland's sequence generate 18321 primes, 3074 of which are distinct. - Giovanni Resta, Apr 08 2016
Same as A137613 with duplicates deleted; same as A132199 with 1s and duplicates deleted. - Jonathan Sondow, May 03 2013
LINKS
Giovanni Resta, Table of n, a(n) for n = 1..3074[Terms 1 to 1000 were computed by T. D. Noe; terms 1001 to 3074 by Giovanni Resta, Apr 08 2016]
Ivars Peterson, A new formula for generating primes
Eric Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8
Wikipedia, Formula for primes
FORMULA
Entries stem from new adjacent differences b(n) = b(n - 1) + GCD(n, b(n - 1)) where b(1)=7.
EXAMPLE
b(5)-b(4) = 15-10 = 5, so a(1)=5.
b(6)-b(5) = 18-15 = 3, so a(2)=3.
b(11)-b(10) = 33-22 =11, so a(3)=11.
MATHEMATICA
t = {}; b1 = 7; Do[b0 = b1; b1 = b0 + GCD[n, b0]; d = b1 - b0; If[d > 1 && !MemberQ[t, d], AppendTo[t, d]], {n, 2, 10^6}]; t (* T. D. Noe, Apr 10 2013 *)
Rest[ DeleteDuplicates[ f[1] = 7; f[n_] := f[n] = f[n - 1] + GCD[n, f[n - 1]]; Differences[ Table[ f[n], {n, 10^6}]]]] (* Jonathan Sondow, May 03 2013 *)
PROG
See the Shallit link for code in Haskell and C.
(Haskell)
import Data.Set (singleton, member, insert)
a221869 n = a221869_list !! (n-1)
a221869_list = f 2 7 (singleton 1) where
f u v s | d `member` s = f (u + 1) (v + d) s
| otherwise = d : f (u + 1) (v + d) (d `insert` s)
where d = gcd u v
-- Reinhard Zumkeller, Nov 15 2013
CROSSREFS
KEYWORD
nonn
AUTHOR
Bill McEachen, Apr 10 2013
EXTENSIONS
More terms from T. D. Noe, Apr 11 2013
Edited by N. J. A. Sloane, Apr 12 2013 at the suggestion of Eric Rowland.
STATUS
approved