|
| |
|
|
A137613
|
|
Omit the 1s from the sequence f(n) - f(n-1) = gcd(n,f(n-1)), where f(1) = 7.
|
|
5
| |
|
|
5, 3, 11, 3, 23, 3, 47, 3, 5, 3, 101, 3, 7, 11, 3, 13, 233, 3, 467, 3, 5, 3, 941, 3, 7, 1889, 3, 3779, 3, 7559, 3, 13, 15131, 3, 53, 3, 7, 30323, 3, 60647, 3, 5, 3, 101, 3, 121403, 3, 242807, 3, 5, 3, 19, 7, 5, 3, 47, 3, 37, 5, 3, 17, 3, 199, 53, 3, 29, 3, 486041, 3, 7, 421, 23
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 1,1
|
|
|
COMMENTS
| Rowland proves that each term is prime. He says it is likely that all odd primes occur.
In the first 5000 terms, there are 965 unique primes and 397 is the least odd prime that does not appear. - T. D. Noe, Mar 01 2008
In the first 10000 terms, the least odd prime that does not appear is 587, according to Rowland. [From Jonathan Sondow, Aug 14 2008]
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n = 1..5000
John Moyer, Source code in C and C++ to print this sequence or sorted and unique values from this sequence. [From John Moyer (jrm(AT)rsok.com), Nov 06 2009]
Ivars Peterson, A New Formula for Generating Primes, The Mathematical Tourist.
Eric S. Rowland, A simple prime-generating recurrence, Abstracts Amer. Math. Soc. 29 (No. 1, 2008), p. 50 (Abstract 1035-11-986). arXiv:0710.3217
Eric S. Rowland, A natural prime-generating recurrence, J. of Integer Sequences 11 (2008), Article 08.2.8.
Eric Rowland, A simple recurrence that produces complex behavior ..., A New Kind of Science blog.
Jeffrey Shallit, Rutgers Graduate Student Finds New Prime-Generating Formula, Recursivity blog.
V. Shevelev, Generalizations of the Rowland theorem
|
|
|
FORMULA
| Denote by Lpf(n) the least prime divisor of n. Then a(n) = Lpf(6-n+sum{i=1,...,n-1}a(i)). [From Vladimir Shevelev, Mar 03 2010]
|
|
|
EXAMPLE
| f(n) = 7, 8, 9, 10, 15, 18, 19, 20, ..., so f(n) - f(n-1) = 1, 1, 1, 5, 3, 1, 1, ... and a(n) = 5, 3, ... .
We have a(1) = Lpf(6-1) = 5; a(2) = Lpf(6-2+5) = 3; a(3) = Lpf(6-3+5+3) = 11; a(4) = Lpf(6-4+5+3+11) = 3; a(5) = Lpf(6-5+5+3+11+3) = 23. [From Vladimir Shevelev, Mar 03 2010]
|
|
|
MAPLE
| A137613_list := proc(n)
local a, c, k, L;
L := NULL; a := 7;
for k from 2 to n do
c := igcd(k, a);
a := a + c;
if c > 1 then L:=L, c fi;
od;
L end:
A137613_list(500000); # Peter Luschny, Nov 17 2011
|
|
|
MATHEMATICA
| f[1] = 7; f[n_] := f[n] = f[n - 1] + GCD[n, f[n - 1]]; DeleteCases[Differences[Table[f[n], {n, 10^6}]], 1] (* Alonso del Arte, Nov 17 2011 *)
|
|
|
CROSSREFS
| f(n) = f(n-1) + gcd(n, f(n-1)) = A106108(n) and f(n) - f(n-1) = A132199(n-1).
Cf. also A084662, A084663, A134734, A134736, A134743, A134744.
Sequence in context: A141620 A195140 A049829 * A165670 A141234 A130180
Adjacent sequences: A137610 A137611 A137612 * A137614 A137615 A137616
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Jan 29 2008, Jan 30 2008
|
| |
|
|
