A137613 Omit the 1's from Rowland's sequence f(n) - f(n-1) = gcd(n,f(n-1)), where f(1) = 7.

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

Offset: 1

Author

Jonathan Sondow, Jan 29 2008, Jan 30 2008

Keywords

• nonn

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 distinct 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. - Jonathan Sondow, Aug 14 2008

Removing duplicates from this sequence yields A221869. The duplicates are A225487. - Jonathan Sondow, May 03 2013

Examples

			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, ... .
From _Vladimir Shevelev_, Mar 03 2010: (Start)
  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. (End)
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..5000
• Jean-Paul Delahaye, Déconcertantes conjectures, Pour la science, 5 (2008), 92-97.
• Brian Hayes, Pumping the Primes, bit-player, 19 August 2015.
• 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).
• Eric S. Rowland, A natural prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.
• 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.
• Eric Rowland, Prime-Generating Recurrence, Wolfram Demonstrations Project, 2008.
• Eric Rowland, A Bizarre Way to Generate Primes, YouTube video, 2023.
• Jeffrey Shallit, Rutgers Graduate Student Finds New Prime-Generating Formula, Recursivity blog.
• Vladimir Shevelev, Generalizations of the Rowland theorem, arXiv:0911.3491 [math.NT], 2009-2010.
• Wikipedia, Formula for primes.

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, A221869.

Cf. A020639, A168008, A190894, A231900.

Programs

Haskell

a137613 n = a137613_list !! (n-1)
a137613_list =  filter (> 1) a132199_list
-- Reinhard Zumkeller, Nov 15 2013

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 *)

PARI

ub=1000; n=3; a=9; while(nDaniel Constantin Mayer, Aug 31 2014

Python

from itertools import count, islice
from math import gcd
def A137613_gen(): # generator of terms
    a = 7
    for n in count(2):
        if (b:=gcd(a,n)) > 1: yield b
        a += b
A137613_list = list(islice(A137613_gen(),20)) # Chai Wah Wu, Mar 14 2023

Formula

Denote by Lpf(n) the least prime factor of n. Then a(n) = Lpf( 6-n+Sum_{i=1..n-1} a(i) ). - Vladimir Shevelev, Mar 03 2010

a(n) = A168008(2*n+4) (conjectured). - Jon Maiga, May 20 2021

a(n) = A020639(A190894(n)). - Seiichi Manyama, Aug 11 2023