A106108 -id:A106108 - cp's OEIS frontend

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

N. J. A. Sloane, Jan 28 2008

nonn

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)

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

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.

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

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

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

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 *)

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

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

A141537 An example of a simple prime-generating algorithm similar to Rowland's (A106108) that is a particular instance of a more general algorithm (see comments).

Original entry on oeis.org

47, 227, 71, 359, 113, 563, 173, 839, 251, 1187, 347, 1607, 461, 2099, 593, 2663, 743, 3299, 911, 4007, 1097, 4787, 1301, 5639, 1523, 6563, 43, 7559, 43, 8627, 2297, 9767, 2591, 10979, 2903, 12263, 53, 13619, 3581, 41, 3947, 16547, 61, 18119, 4733, 19763, 5153, 47

Offset: 1

Aldrich Stevens (aldrichstevens(AT)msn.com), Aug 15 2008

nonn

Below is a general algorithm that can be used as a starting point for finding similar ones and three examples.
Not every possibility will work (additional conditions may apply) but it is easy to see that there are an infinite number of algorithms much like Rowland's that will have hundreds or thousands of primes between the 1's before a composite is encountered.
1) Initialize the integers x, k, a and b and choose f(x), g(k).
2) Repeat indefinitely:
   2a) x = x + 1;
   2b) set c = GCD( f(x), f(x - 1) - a*g(k) );
   2c) if c > 1, then c is a term of the sequence and k = k + b.
The present sequence is generated by using f(x) = x^2 - x + 41, g(k) = k, x = 1, k = 2, a = 3 and b = 1.
Examples:
A) f(x) = 5*x^2 + 5*x + 1, g(k) = k, x = 1, k = 2, a = 10, b = 1. These values generate the sequence: 11, 31, 61, 101, 151, 211, 281, 19, 41, 29, 661, 11, 911, 1051, 1201, 1361, 1531, 59, 1901, ...
B) f(x) = x^2 - x + 41, g(k) = k, x = 1, k = 2, a = 3, b = 1. These values generate the sequence: 47, 227, 71, 359, 113, 563, 173, 839, 251,1187,347, 1607, 461,2099,593, 2663, 743,3299, 911, 4007, ...
C) f(x) = 5*x^2 + 5*x + 1, g(k) = k^2 - k + 41, x = 1, k = 2, a = 2, b = 1. These values generate the sequence: 11, 1979, 2549, 11,4691, 11, 8929, 29, 11, 22051, 41, 19, 48619, 61751, 11, 229, 11, 144779, 175141, 11, ...

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

Paolo Xausa, Table of n, a(n) for n = 1..10000

Cf. A084662, A106108, A137613.

Module[{k = 2, c, f}, f[x_] := x^2 - x + 41; Table[If[(c = GCD[f[x], f[x - 1] - 3*k]) > 1, k++; c, Nothing], {x, 12000}]] (* Paolo Xausa, Jan 31 2025 *)

Edited by Paolo Xausa, Jan 31 2025

A006992 Bertrand primes: a(n) is largest prime < 2*a(n-1) for n > 1, with a(1) = 2.

Original entry on oeis.org

2, 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259, 2503, 5003, 9973, 19937, 39869, 79699, 159389, 318751, 637499, 1274989, 2549951, 5099893, 10199767, 20399531, 40799041, 81598067, 163196129, 326392249, 652784471, 1305568919, 2611137817

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

a(n) < a(n+1) by Bertrand's postulate (Chebyshev's theorem). - Jonathan Sondow, May 31 2014
Let b(n) = 2^n - a(n). Then b(n) >= 2^(n-1) - 1 and b(n) is a B_2 sequence: 0, 1, 3, 9, 19, 41, 85, 173, 349, ... - Thomas Ordowski, Sep 23 2014 See the link for B_2 sequence.
These primes can be obtained of exclusive form using a restricted variant of Rowland's prime-generating recurrence (A106108), making gcd(n, a(n-1)) = -1 when GCDs are greater than 1 and less than n (see program). These GCDs are also a divisor of each odd number from a(n) + 2 to 2*a(n-1) - 1 in reverse order, so that this subtraction with -1's invariably leads to the prime. - Manuel Valdivia, Jan 13 2015
First row of array in A229607. - Robert Israel, Mar 31 2015
Named after the French mathematician Joseph Bertrand (1822-1900). - Amiram Eldar, Jun 10 2021

Martin Aigner and Günter M. Ziegler, Proofs from The Book, Springer-Verlag, Berlin, 1999; see p. 7.
Martin Griffiths, The Backbone of Pascal's Triangle, United Kingdom Mathematics Trust (2008), page 115. [From Martin Griffiths, Mar 28 2009]
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 344.
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers. 2nd ed., Wiley, NY, 1966, p. 189.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Robert G. Wilson v, Table of n, a(n) for n = 1..1001 (first 100 terms from T. D. Noe)
Paul Erdős, Beweis eines Satzes von Tschebyschef (in German), Acta Litt. Sci. Szeged, Vol. 5 (1932), pp. 194-198.
Paul Erdős, A theorem of Sylvester and Schur, J. London Math. Soc., Vol. 9 (1934), pp. 282-288.
Srinivasa Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., Vol. 11 (1919), pp. 181-182.
Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq., Vol. 15 (2012) Article 12.5.4.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, arXiv:0907.5232 [math.NT], 2009-2010.
Jonathan Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly, Vol. 116, No. 7 (2009), pp. 630-635.
Jonathan Sondow and Eric Weisstein, MathWorld: Bertrand's Postulate.
Eric Weisstein's World of Mathematics, B2 Sequence.
Robert G. Wilson, V, Letter to N. J. A. Sloane, Oct. 1993.

Cf. A055496, A163961.
See A185231 for another version.
Cf. A007917, A229607, A295262.

a006992 n = a006992_list !! (n-1)
a006992_list = iterate (a007917 . (* 2)) 2
-- Reinhard Zumkeller, Sep 17 2014

A006992 := proc(n) option remember; if n=1 then 2 else prevprime(2*A006992(n-1)); fi; end;

bertrandPrime[1] = 2; bertrandPrime[n_] := NextPrime[ 2*a[n - 1], -1]; Table[bertrandPrime[n], {n, 40}]
(* Second program: *)
NestList[NextPrime[2#, -1] &, 2, 40] (* Harvey P. Dale, May 21 2012 *)
k = 3; a[n_] := If[GCD[n,k] > 1 && GCD[n, k] < n, -1, GCD[n, k]]; Select[Differences@Table[k = a[n] + k, {n, 2611137817}], # > 1 &] (* Manuel Valdivia, Jan 13 2015 *)

print1(t=2);for(i=2,60,print1(", "t=precprime(2*t))) \\ Charles R Greathouse IV, Apr 01 2013

from sympy import prevprime
l = [2]
for i in range(1, 51):
    l.append(prevprime(2 * l[i - 1]))
print(l) # Indranil Ghosh, Apr 26 2017

a(n+1) = A007917(2*a(n)). - Reinhard Zumkeller, Sep 17 2014

Limit_{n -> infinity} a(n)/2^n = 0.303976447924... - Thomas Ordowski, Apr 05 2015

Definition completed by Jonathan Sondow, May 31 2014

B_2 sequence link added by Wolfdieter Lang, Oct 09 2014

A135506 a(n) = x(n+1)/x(n) - 1 where x(1)=1 and x(k) = x(k-1) + lcm(x(k-1),k). Here x(n) = A135504(n).

Original entry on oeis.org

2, 1, 2, 5, 1, 7, 1, 1, 5, 11, 1, 13, 1, 5, 1, 17, 1, 19, 1, 1, 11, 23, 1, 5, 13, 1, 1, 29, 1, 31, 1, 11, 17, 1, 1, 37, 1, 13, 1, 41, 1, 43, 1, 1, 23, 47, 1, 1, 1, 17, 13, 53, 1, 1, 1, 1, 29, 59, 1, 61, 1, 1, 1, 13, 1, 67, 1, 23, 1, 71, 1, 73, 1, 1, 1, 1, 13, 79, 1, 1, 41, 83, 1, 1, 43, 29, 1, 89

Offset: 1

Benoit Cloitre, Feb 09 2008

nonn

This sequence has properties related to primes. For instance: terms consist of 1's or primes only; if 3 never occurs, any prime p occurs finitely many times.
All prime numbers 'p' from the sequence A014963(n), which equals A003418(n+1)/A003418(n), are in a(n-1) = p. - Eric Desbiaux, Jan 11 2015
For any prime p > 3, a(p-1) = p. Also a(n) is not 3 for any n. All terms but a(1) and a(3) are odd, and probably all of them are not composite numbers; this is strongly related to a strong version of Linnik's Theorem (see Ruiz-Cabello link). - Serafín Ruiz-Cabello, Sep 30 2015
Per the prior comment, the distinct prime terms correspond to A045344. This is every prime except for 3. - Bill McEachen, Sep 12 2022

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 10000 terms from Robert Israel)
Eric S. Rowland, Prime-Generating Recurrences and a Tale of Logarithmic Scale, YouTube video, 2023. (See especially the last section beginning at 20:08).
Serafín Ruiz-Cabello, On the use of the lowest common multiple to build a prime-generating recurrence, arXiv:1504.05041 [math.CO], 2015.

Cf. A045344, A135504, A361460, A361461 (positions of 1's), A361462 [= a(n) mod 4], A361463, A361464, A361470.

Cf. also A106108.

x[1]:= 1;
for n from 2 to 101 do
  x[n]:= x[n-1] + ilcm(x[n-1],n);
  a[n-1]:= x[n]/x[n-1]-1;
od:
seq(a[n],n=1..100); # Robert Israel, Jan 11 2015

a[n_] := x[n+1]/x[n] - 1; x[1] = 1; x[k_] := x[k] = x[k-1] + LCM[x[k-1], k]; Table[a[n], {n, 1, 88}] (* Jean-François Alcover, Jan 08 2013 *)

x1=1;for(n=2,40,x2=x1+lcm(x1,n);t=x1;x1=x2;print1(x2/t-1,","))

from itertools import count, islice
from math import lcm
def A135506_gen(): # generator of terms
    x = 1
    for n in count(2):
        y = x+lcm(x,n)
        yield y//x-1
        x = y
A135506_list = list(islice(A135506_gen(),20)) # Chai Wah Wu, Mar 13 2023

a(n) = A135504(n+1)/A135504(n) - 1.

a(n) = (n+1) / A361470(n). - Antti Karttunen, Mar 26 2023

References to A135504 added by Antti Karttunen, Mar 07 2023

A084662 a(1) = 4; a(n) = a(n-1) + gcd(a(n-1), n).

Original entry on oeis.org

4, 6, 9, 10, 15, 18, 19, 20, 21, 22, 33, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 69, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 141, 144, 145, 150, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168

Offset: 1

Matthew Frank (mfrank(AT)wopr.wolfram.com) on behalf of the 2003 New Kind of Science Summer School, Jul 15 2003

nonn

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

Indranil Ghosh, Table of n, a(n) for n = 1..50000 (terms 1..1000 from T. D. Noe)
Eric S. Rowland, A simple prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.

Cf. A084663, A106108 and other sequences mentioned in A106108.

Cf. A134734 (first differences), A134736, A230504.

a084662 n = a084662_list !! (n-1)
a084662_list =
   4 : zipWith (+) a084662_list (zipWith gcd a084662_list [2..])
-- Reinhard Zumkeller, Nov 15 2013

[n eq 1 select 4 else Self(n-1)+Gcd(Self(n-1),n): n in [1..66]]; // Bruno Berselli, May 24 2011

S := 4; f := proc(n) option remember; global S; if n=1 then S else f(n-1)+igcd(n,f(n-1)); fi; end;

a[1]= 4; a[n_]:= a[n]= a[n-1] + GCD[n, a[n-1]]; Table[a[n], {n, 70}]
nxt[{n_, a_}]:= {n+1, a + GCD[a, n+1]}; NestList[nxt,{1,4},70][[All,2]] (* Harvey P. Dale, Dec 25 2018 *)

a[1]:4$ a[n]:=a[n-1]+gcd(a[n-1],n)$ makelist(a[n], n, 1, 66); /* Bruno Berselli, May 24 2011 */

@CachedFunction
def a(n): # a = A084662
    if (n==1): return 4
    else: return a(n-1) + gcd(a(n-1), n)
[a(n) for n in range(1,71)] # G. C. Greubel, Mar 22 2023

A084663 a(1) = 8; a(n) = a(n-1) + gcd(a(n-1), n).

Original entry on oeis.org

8, 10, 11, 12, 13, 14, 21, 22, 23, 24, 25, 26, 39, 40, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 177, 180, 181, 182, 189, 190

Offset: 1

Matthew Frank (mfrank(AT)wopr.wolfram.com) on behalf of the 2003 New Kind of Science Summer School, Jul 15 2003

nonn

The first 150000000 differences are all primes or 1. Is this true in general?

The proof of the conjecture is identical to the proof in the Rowland link. - Yifan Xie, Apr 11 2025

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

Indranil Ghosh, Table of n, a(n) for n = 1..50000
Eric S. Rowland, A natural prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.
Eric S. Rowland, A natural prime-generating recurrence, JIS 11 (2008) 08.2.8.

Cf. A084662, A106108.

Cf. A230504, A134744 (first differences), A134736.

a084663 n = a084663_list !! (n-1)
a084663_list =
   8 : zipWith (+) a084663_list (zipWith gcd a084663_list [2..])
-- Reinhard Zumkeller, Nov 15 2013

S := 8; f := proc(n) option remember; global S; if n=1 then S else f(n-1)+igcd(n,f(n-1)); fi; end;

a[n_]:= a[n]= If[n==1,8, a[n-1] + GCD[n, a[n-1]]]; Table[a[n], {n,70}]
RecurrenceTable[{a[1]==8,a[n]==a[n-1]+GCD[a[n-1],n]},a,{n,70}] (* Harvey P. Dale, Apr 12 2016 *)

@CachedFunction
def a(n): # a = A084663
    if (n==1): return 8
    else: return a(n-1) + gcd(a(n-1), n)
[a(n) for n in range(1, 71)] # G. C. Greubel, Mar 22 2023

A186253 Indices of zeros of the sequence u(n)=abs(u(n-1)-gcd(u(n-1),n-1)), u(1)=1.

Original entry on oeis.org

2, 5, 11, 23, 47, 79, 157, 313, 619, 1237, 2473, 4909, 9817, 19603, 39199, 78193, 156019, 311347, 622669, 1244149, 2487739, 4975111, 9950221, 19900399, 39800797, 79601461, 159202369, 318404629, 636788881, 1273577761, 2547155419, 5094310069, 10188620041

Offset: 1

Benoit Cloitre, Feb 16 2011

nonn

For any fixed integer m>=1 define u(1)=1 and u(n)=abs(u(n-1)-gcd(u(n-1),m*n-1)). Then (b_m(k))_{k>=1} is the sequence of integers such that u(b_m(k))=0 and we conjecture that for k large enough m*b_m(k)+m-1 is a prime number. Here for m=1 it appears a(n) is prime for n>=1.
See A261301 for the sequence u relevant here (m=1). - M. F. Hasler, Aug 14 2015
A261301(a(n)-1) = 1; A261301(a(n)) = 0; A261301(a(n)+1) = a(n). - Reinhard Zumkeller, Sep 07 2015

Moritz Firsching, Table of n, a(n) for n = 1..315
B. Cloitre, 10 conjectures in additive number theory, arXiv:1101.4274 [math.NT], 2011.
M. F. Hasler, Rowland-Cloître type prime generating sequences, OEIS Wiki, August 2015.

Cf. A106108.

Cf. A261301 - A261310; A186254 - A186263.

a186253 n = a186253_list !! (n-1)
a186253_list = filter ((== 0) . a261301) [1..]
-- Reinhard Zumkeller, Sep 07 2015

a = m = 1; Reap[For[n = 2, n <= 10^7, n++, a = Abs[a - GCD[a, m*n - 1]]; If[a == 0, Print[m*n + m - 1]; Sow[m*n + m - 1]]]][[2, 1]] (* Jean-François Alcover, Feb 05 2019, from PARI *)
nxt[{n_,a_}]:={n+1,Abs[a-GCD[a,n]]}; Position[NestList[nxt,{1,1},13*10^5][[All,2]],0]// Flatten (* The program generates the first 20 terms of the sequence. *) (* Harvey P. Dale, Oct 02 2022 *)

a=1;m=1;for(n=2,1e7,a=abs(a-gcd(a,m*n-1));if(a==0,print1(m*n+m-1,",")))

next_a(last_a) = {
  local(A=last_a,B=last_a,C=2*last_a+1);
  while(A>0,
    D=divisors(C);
    k1=10*D[2];
    for(j=2,#D, d=D[j];k=((A+1-B+d)/2)%d;
      if(k==0,k=d); if(k<=k1,k1=k;d1=d));
    if(k1-1+d1==A,B=B+1);
    A = max(A-(k1-1)-d1,0);
    B = B + k1;
    C = C - (d1 - 1);
  );
  return(B);
}
a=2
for(n=1,99,print1(a,", ");a=next_a(a)) \\ Jan Büthe and Moritz Firsching, Aug 04 2015

m=a=k=1; for(n=1, 30, while( a>d=vecmin(apply(p->a%p, factor(N=m*(k+a)+m-1)[,1])), a-=d+gcd(a-d,N); k+=1+d); k+=a+1; print1(a=N,",")) \\ M. F. Hasler, Aug 22 2015

Conjecture: a(n) is asymptotic to c*2^n with c = 1.1861...

Definition clarified by M. F. Hasler, Aug 14 2015

A261301 a(n+1) = abs(a(n) - gcd(a(n), n)), a(1) = 1.

Original entry on oeis.org

1, 0, 2, 1, 0, 5, 4, 3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 47, 46, 45, 40, 39, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 79, 78, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60

Offset: 1

M. F. Hasler, Aug 14 2015

nonn

The absolute value is relevant only when a(n) = 0 in which case a(n+1) = gcd(a(n),n) = n.
It is conjectured that a(n) = 0 implies that n is prime, see A186253. (This is the sequence {u(n)} mentioned there.)
a(A186253(n)-1) = 1; a(A186253(n)) = 0; a(A186253(n)+1) = A186253(n). - Reinhard Zumkeller, Sep 07 2015

			a(2) = a(1) - gcd(a(1),1) = 1 - 1 = 0.
a(3) = |a(2) - gcd(a(2),2)| = gcd(0,2) = 2 is prime.
a(3+2) = a(5) = 0, a(6)  = gcd(0,5) = 5 is prime.
a(6+5) = a(11) = 0, a(12) = gcd(0,11) = 11 is prime.
a(12+11) = a(23) = 0, a(24) = 23 is prime.
a(24+23) = a(47) = 0, a(48) = 47 is prime.
a(50) = 45 and gcd(45,50) = 5, thus a(51) = 45 - 5 = 40.
a(52) = 39 and gcd(39,52) = 13, thus a(53) = 39 - 13 = 26. Then, a(53+26) = 0 and 79 = a(80) is prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A186253 - A186263, A261302 - A261310, A106108.

a261301 n = a261301_list !! (n-1)
a261301_list = 1 : map abs
   (zipWith (-) a261301_list (zipWith gcd [1..] a261301_list))
-- Reinhard Zumkeller, Sep 07 2015

FoldList[Abs[#1-GCD[#1,#2]]&,1,Range@96] (* Ivan N. Ianakiev, Aug 15 2015 *)

print1(a=1);m=1;for(n=1,199, print1( ",",a=abs(a-gcd(a,n))))

A186263 a(n) = 10*b_10(n) + 9, where b_10 lists the indices of zeros of the sequence A261310: u(n) = abs(u(n-1) - gcd(u(n-1), 10n-1)), u(1) = 1.

Original entry on oeis.org

29, 269, 2969, 32609, 357169, 3928669, 43213789, 475113649, 5226205969, 57488152069, 632360271769, 6955957188049, 76515529068529, 841670819753809, 9258379017291889, 101842168949117209, 1120263858440288929, 12322902442843176229, 135551926871245562989

Offset: 1

Benoit Cloitre, Feb 16 2011

nonn

For any fixed integer m>=1 define u(1)=1 and u(n)=abs(u(n-1)-gcd(u(n-1),m*n-1)). Then (b_m(k))_{k>=1} is the sequence of integers such that u(b_m(k))=0 and we conjecture that for k large enough m*b_m(k)+m-1 is a prime number. Here for m=10 it appears a(n) is prime for n>=1.

See A261310 for the sequence u relevant here (m=10). - M. F. Hasler, Aug 14 2015

Benoit Cloitre, 10 conjectures in additive number theory, preprint arXiv:2011.4274 [math.NT] , 2011.
M. F. Hasler, Rowland-Cloître type prime generating sequences, OEIS Wiki, August 2015.

Cf. A106108.

Cf. A261301 - A261310; A186253 - A186261.

a=1; m=10; for(n=2, 1e7, a=abs(a-gcd(a, m*n-1)); if(a==0, print1(m*n+m-1, ", ")))

m=10; a=k=1; for(n=1, 20, while( a>D=vecmin(apply(p->a%p, factor(N=m*(k+a)+m-1)[, 1])), a-=D+gcd(a-D, N); k+=1+D); k+=a+1; print1(a=N, ", ")) \\ M. F. Hasler, Aug 22 2015

We conjecture that a(n) is asymptotic to c*11^n with c>0.

See the wiki link for a sketch of a proof of this conjecture. We find c = 2.2163823215... - M. F. Hasler, Aug 22 2015

Edited by M. F. Hasler, Aug 14 2015

More terms from M. F. Hasler, Aug 22 2015

A134162 Let S(k) be the sequence s() defined by s(1) = k; for i > 1, s(i) = s(i-1) + gcd(s(i-1), i). Start with the list of positive integers and remove any k's for which S(k) merges with an S(m) with m < k. Each value k > 1 is conjectural.

Original entry on oeis.org

1, 2, 4, 8, 16, 20, 44, 92, 110, 136, 152, 170, 172, 188, 200, 212, 236, 242, 256, 272, 316, 332, 368, 440, 488, 500, 590, 616, 620, 632, 650, 676, 704, 710, 742, 788, 824, 848, 892, 946, 952, 968, 1010, 1034, 1036, 1052, 1058, 1088, 1118

Offset: 1

Eric Rowland, Jan 29 2008

nonn

For the given initial values k, it is conjectural that their sequences S(k) never merge. The S(k) have been checked to be distinct for 2^60 terms (see Rowland link), but it is possible that they merge later on.

			From _Danny Rorabaugh_, Apr 02 2015: (Start)
S(1) = A000027 is the positive integers.
S(2) = [2,4,5,...,i+2,...].
S(3) = [3,4,5,...,i+2,...] merges with S(2) at index 2.
S(4) = [4,6,9,10,15,18,19,20,21,22,33,...] = A084662.
S(5) = [5,6,9,...] = A134736 merges with S(4) at index 2.
(End)

Eric Rowland, A Natural Prime-Generating Recurrence (Section 5: Primes), Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8.

Cf. A000027 (S(1)), A084662 (S(4)), A134736 (S(5)), A106108 (S(7)), A084663 (S(8)).

Cf. A106108 for other Crossrefs.

cp's OEIS Frontend

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

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A141537 An example of a simple prime-generating algorithm similar to Rowland's (A106108) that is a particular instance of a more general algorithm (see comments).

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A006992 Bertrand primes: a(n) is largest prime < 2*a(n-1) for n > 1, with a(1) = 2.

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A135506 a(n) = x(n+1)/x(n) - 1 where x(1)=1 and x(k) = x(k-1) + lcm(x(k-1),k). Here x(n) = A135504(n).

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A084662 a(1) = 4; a(n) = a(n-1) + gcd(a(n-1), n).

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A084663 a(1) = 8; a(n) = a(n-1) + gcd(a(n-1), n).

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