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A137613 Omit the 1s from Rowland's sequence f(n) - f(n-1) = gcd(n,f(n-1)), where f(1) = 7. 10
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; text; 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. [Jonathan Sondow, Aug 14 2008]

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..5000

Jean-Paul Delahaye, D├ęconcertantes conjectures, Pour la science, 5 (2008), 92-97. [broken link]

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). 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.

Eric Rowland, Prime-Generating Recurrence, Wolfram Demonstrations Project, 2008.

Jeffrey Shallit, Rutgers Graduate Student Finds New Prime-Generating Formula, Recursivity blog.

V. Shevelev, Generalizations of the Rowland theorem, arXiv 2009

Wikipedia, Formula for primes

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]

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. [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 *)

PROG

(Haskell)

a137613 n = a137613_list !! (n-1)

a137613_list =  filter (> 1) a132199_list

-- Reinhard Zumkeller, Nov 15 2013

(PARI)

ub=1000; n=3; a=9; while(n<ub, m=a\n; d=factor((m-1)*n-1)[1, 1]; print1(d, ", "); n=n+((d-1)\(m-1)); a=m*n; ); \\ Daniel Constantin Mayer, Aug 31 2014

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. A231900.

Sequence in context: A195140 A049829 A258333 * A259650 A165670 A221869

Adjacent sequences:  A137610 A137611 A137612 * A137614 A137615 A137616

KEYWORD

nonn

AUTHOR

Jonathan Sondow, Jan 29 2008, Jan 30 2008

STATUS

approved

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Last modified November 20 13:59 EST 2017. Contains 294972 sequences.