A132199 Rowland's prime-generating sequence: first differences of A106108.

1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 23, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 47, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 101, 3, 1, 1

Offset: 1

Author

N. J. A. Sloane, Jan 28 2008

Keywords

• nonn

Comments

Rowland shows that the terms are all 1's or primes.

The prime terms form A137613.

See A137613 for additional comments, links and references. - Jonathan Sondow, Aug 14 2008

From Robert G. Wilson v, Apr 30 2009: (Start)

First appearance of k-th prime, k >= 0: 1, 0, 5, 4, 104, 10, 116, 242878, 242819, 22, 243019, 3891770, 242867, ..., .

The number of different numbers in the first 10^k terms beginning with k=0: 1, 4, 7, 12, 15, 19, 30, >35, ..., .

Record high values are A191304. (End)

References

• Eric S. Rowland, A simple prime-generating recurrence, Abstracts Amer. Math. Soc., 29 (No. 1, 2008), p. 50 (Abstract 1035-11-986).

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000 .
• Jean-Paul Delahaye, Déconcertantes conjectures, Pour la Science (French edition of Scientific American), No. 367, May 2008.
• Brian Hayes, Pumping the Primes, bit-player, 19 August 2015.
• Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 18.
• Eric S. Rowland, A natural prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.
• Eric S. Rowland, A simple prime-generating recurrence, 2008.
• Eric S. Rowland, Prime-Generating Recurrences and a Tale of Logarithmic Scale, 20 January 2023.

Crossrefs

Cf. A106108, A137613, A134734, A134743, A134744, A191304 (record highs) A247090. See A106108 for other cross-references.

Programs

Haskell

a132199 n = a132199_list !! (n-1)
a132199_list = zipWith (-) (tail a106108_list) a106108_list
-- Reinhard Zumkeller, Nov 15 2013

Maple

A106108 := proc(n)
    option remember;
    if n = 1 then
        7;
    else
        procname(n-1)+igcd(n,procname(n-1)) ;
    end if;
end proc:
A132199 := proc(n)
    A106108(n+1)-A106108(n) ;
end proc: # R. J. Mathar, Jul 04 2013

Mathematica

a[1] = 7; a[n_] := a[n] = a[n - 1] + GCD[n, a[n - 1]]; t = Array[a, 104]; Rest@t - Most@t (* Robert G. Wilson v, Apr 30 2009 *)

PARI

ub=1000; a=7; n=2; while(nDaniel Constantin Mayer, Aug 31 2014

Python

from itertools import count, islice
from math import gcd
def A132199_gen(): # generator of terms
    a = 7
    for n in count(2):
        yield (b:=gcd(a,n))
        a += b
A132199_list = list(islice(A132199_gen(),20)) # Chai Wah Wu, Mar 14 2023