A137613 -id:A137613 - cp's OEIS frontend

A226781 Number of 1's in A132199 preceding the n-th Rowland prime, A137613(n).

3, 3, 7, 7, 17, 17, 39, 39, 40, 40, 89, 89, 91, 95, 95, 100, 215, 215, 447, 447, 448, 448, 917, 917, 919, 1862, 1862, 3750, 3750, 7528, 7528, 7533, 15097, 15097, 15122, 15122, 15124, 30284, 30284, 60606, 60606, 60607, 60607, 60656, 60656, 121356, 121356

Offset: 1

Vladimir Shevelev, Jun 29 2013

nonn

The length of the clusters of 1's in A132199 is 3, 0, 4, 0, 10, 0, 22, 0, 1, 0, 49,.. and this sequence here are the partial sums of these lengths.

Cf. A132199, A137613.

cd1 := 0 ;
for i from 1 do
    if A132199(i) = 1 then
        cd1 := cd1+1 ;
    else
        printf("%d,\n",cd1) ;
    end if;
end do: # R. J. Mathar, Jul 04 2013

A106108 Rowland's prime-generating sequence: a(1) = 7; for n > 1, a(n) = a(n-1) + gcd(n, a(n-1)).

Original entry on oeis.org

7, 8, 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

N. J. A. Sloane, Jan 28 2008

nonn

The title refers to the sequence of first differences, A132199.
Setting a(1) = 4 gives A084662.
Rowland proves that the first differences are all 1's or primes. The prime differences form A137613.
See A137613 for additional comments, links and references. - Jonathan Sondow, Aug 14 2008
Not all starting values generate differences of all 1's or primes. The following a(1) generate composite differences: 532, 533, 534, 535, 698, 699, 706, 707, 708, 709, 712, 713, 714, 715, ... - Dmitry Kamenetsky, Jul 18 2015
The same results are obtained if 2's are removed from n when gcd is performed, so the following is also true: a(1) = 7; for n > 1, a(n) = a(n-1) + gcd(A000265(n), a(n-1)). - David Morales Marciel, Sep 14 2016

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..25000 (terms 1..1000 from T. D. Noe)
Fernando Chamizo, Dulcinea Raboso and Serafin Ruiz-Cabello, On Rowland's sequence, Electronic J. Combin., Vol. 18(2), 2011, #P10.
Brian Hayes, Pumping the Primes, bit-player, 19 August 2015.
Eric S. Rowland, A simple prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.
Eric S. Rowland, Prime-Generating Recurrence, Wolfram Demonstrations Project. - _Robert G. Wilson v_, Sep 10 2008
Eric S. Rowland, Prime-Generating Recurrences and a Tale of Logarithmic Scale, YouTube video, 2023.

Cf. A084662, A084663, A132199, A134734, A134736, A134743, A134744, A134162, A137613, A221869.

Cf. A230504.

a106108 n = a106108_list !! (n-1)
a106108_list =
   7 : zipWith (+) a106108_list (zipWith gcd a106108_list [2..])
-- Reinhard Zumkeller, Nov 15 2013

[n le 1 select 7 else Self(n-1) + Gcd(n, Self(n-1)): n in [1..70]]; // Vincenzo Librandi, Jul 19 2015

S:=7; f:= proc(n) option remember; global S; if n=1 then RETURN(S); else RETURN(f(n-1)+gcd(n,f(n-1))); fi; end; [seq(f(n),n=1..200)];

a[1] = 7; a[n_] := a[n] = a[n - 1] + GCD[n, a[n - 1]]; Array[a, 66] (* Robert G. Wilson v, Sep 10 2008 *)

a=vector(100);a[1]=7;for(n=2,#a,a[n]=a[n-1]+gcd(n,a[n-1]));a \\ Charles R Greathouse IV, Jul 15 2011

from itertools import count, islice
from math import gcd
def A106108_gen(): # generator of terms
    yield (a:=7)
    for n in count(2):
        yield (a:=a+gcd(a,n))
A106108_list = list(islice(A106108_gen(),20)) # Chai Wah Wu, Mar 14 2023

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

Original entry on oeis.org

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

A190894 Auxiliary c(n) sequence used to prove some properties about Rowland's sequence. c(n) has the following recursive definition: c(1) = 5, c_(n+1) = c(n) + lfp(c(n)) - 1, where lpf(.) denotes the lowest prime factor of a number.

Original entry on oeis.org

5, 9, 11, 21, 23, 45, 47, 93, 95, 99, 101, 201, 203, 209, 219, 221, 233, 465, 467, 933, 935, 939, 941, 1881, 1883, 1889, 3777, 3779, 7557, 7559, 15117, 15119, 15131, 30261, 30263, 30315, 30317, 30323, 60645, 60647, 121293, 121295, 121299, 121301, 121401

Offset: 1

Serafín Ruiz-Cabello, May 23 2011

nonn

This sequence is matched with r(n)=A190895(n). Rowland's sequence (A106108) can be easily described in terms of c(n) and r(n). Also, they can be used to prove easily that the difference between two consecutive terms is always 1 or a prime.
This sequence is related to Rowland's sequence (A106108) with initial condition a(1)=7. For any other odd initial condition a(1) greater than 3, there is an analog c(n) sequence, with c(1) = a(1) - 2.
Sequence r(n) satisfies 2r(n) - 1 = c(n), for any n>1.
For further information, see the references.

			For n=2, c(n) = 5 + lpf(5) - 1 = 5 + 5 - 1 = 9
For n=3, c(n) = 9 + lfp(9) - 1 = 9 + 3 - 1 = 11

Harvey P. Dale, Table of n, a(n) for n = 1..1000
F. Chamizo, D. Raboso, and S. Ruiz-Cabello, On Rowland's sequence, Vol. 18(2), 2011, #P10.
E. S. Rowland, A natural prime-generating recurrence, J. Integer Seq., 11(2): Article 08.2.8, 13, 2008.
Eric Rowland, A Bizarre Way to Generate Primes, YouTube video, 2023.

Cf. A020639, A106108, A137613, A190895.

NestList[#+FactorInteger[#][[1,1]]-1&,5,50] (* Harvey P. Dale, Jun 10 2016 *)

c(1) = 5; c(n+1) = c(n) + lfp(c(n)) - 1.

A221869 New primes found by Rowland's recurrence in the order of their appearance.

Original entry on oeis.org

5, 3, 11, 23, 47, 101, 7, 13, 233, 467, 941, 1889, 3779, 7559, 15131, 53, 30323, 60647, 121403, 242807, 19, 37, 17, 199, 29, 486041, 421, 972533, 577, 1945649, 163, 3891467, 127, 443, 31, 7783541, 15567089, 5323, 31139561, 41, 62279171, 83, 1103, 124559609

Offset: 1

Bill McEachen, Apr 10 2013

nonn

The terms up to 1103 required examining numbers produced by Rowland's recurrence up to n = 10^8. - T. D. Noe, Apr 11 2013
Exactly 177789368686545736460055960459780707068552048703463291 iterations to find the first 1000 terms of this sequence. - T. D. Noe, Apr 13 2013
The first 10^100 terms of Rowland's sequence generate 18321 primes, 3074 of which are distinct. - Giovanni Resta, Apr 08 2016
Same as A137613 with duplicates deleted; same as A132199 with 1s and duplicates deleted. - Jonathan Sondow, May 03 2013

			b(5)-b(4) = 15-10 = 5, so a(1)=5.
b(6)-b(5) = 18-15 = 3, so a(2)=3.
b(11)-b(10) = 33-22 =11, so a(3)=11.

Giovanni Resta, Table of n, a(n) for n = 1..3074[Terms 1 to 1000 were computed by T. D. Noe; terms 1001 to 3074 by _Giovanni Resta_, Apr 08 2016]
Ivars Peterson, A new formula for generating primes
Eric Rowland, A simple recurrence that produces complex behavior — and primes!
Eric Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8
Jeffrey Shallit, Recursivity: Rutgers graduate student finds new prime-generating formula
Wikipedia, Formula for primes

Cf. A132199, A137613.

Cf. A106108.

import Data.Set (singleton, member, insert)
a221869 n = a221869_list !! (n-1)
a221869_list = f 2 7 (singleton 1) where
   f u v s | d `member` s = f (u + 1) (v + d) s
           | otherwise    = d : f (u + 1) (v + d) (d `insert` s)
           where d = gcd u v
-- Reinhard Zumkeller, Nov 15 2013

t = {}; b1 = 7; Do[b0 = b1; b1 = b0 + GCD[n, b0]; d = b1 - b0; If[d > 1 && !MemberQ[t, d], AppendTo[t, d]], {n, 2, 10^6}]; t (* T. D. Noe, Apr 10 2013 *)
Rest[ DeleteDuplicates[ f[1] = 7; f[n_] := f[n] = f[n - 1] + GCD[n, f[n - 1]]; Differences[ Table[ f[n], {n, 10^6}]]]] (* Jonathan Sondow, May 03 2013 *)

Entries stem from new adjacent differences b(n) = b(n - 1) + GCD(n, b(n - 1)) where b(1)=7.

More terms from T. D. Noe, Apr 11 2013

Edited by N. J. A. Sloane, Apr 12 2013 at the suggestion of Eric Rowland.

A191304 Records in A132199.

Original entry on oeis.org

1, 5, 11, 23, 47, 101, 233, 467, 941, 1889, 3779, 7559, 15131, 30323, 60647, 121403, 242807, 486041, 972533, 1945649, 3891467, 7783541, 15567089, 31139561, 62279171, 124559609, 249120239, 498270791, 996541661, 1993083437, 3986167223, 7972334723, 15944673761

Offset: 1

Robert G. Wilson v, Apr 30 2009

nonn

Except for a(1)=1, log(a(n)) is nearly but not exactly linear. - Ralf Steiner, Aug 18 2017

Ralf Steiner, Table of n, a(n) for n = 1..171

Cf. A106108, A132199, A137613.

Union@ FoldList[Max, Rest@ # - Most@ #] &@ FoldList[#1 + GCD[#2, #1] &, 7, Range[2, 10^7]] (* Michael De Vlieger, Aug 19 2017, after Robert G. Wilson v at A132199 *)

For all a(n) except a(1)=1 is 5 congruent mod 6, a(n+1) > 2 a(n) and lim_{n->inf} a(n+1)/a(n) = 2. - Ralf Steiner, Aug 18 2017

a(23)-a(33) from Ralf Steiner, Aug 18 2017

A225487 Duplicate primes found by Rowland's recurrence in the order of their reappearance.

Original entry on oeis.org

3, 5, 11, 7, 13, 101, 47, 53, 23, 19, 29, 37, 31, 41, 83, 73, 17, 43, 67, 157, 179, 167, 79, 443, 139, 113, 137, 97, 233, 61, 823, 71, 103, 151, 199, 499, 181, 229, 353, 313, 1889, 271, 317, 197, 613, 607, 127, 257, 89, 367, 223, 433, 239, 911, 109, 107, 557

Offset: 1

Jonathan Sondow, May 08 2013

nonn

Among the first 10^8 terms of A132199 (Rowland's sequence of 1s and primes), 121 terms are prime. Eleven of them appear more than once, and so are a(1), ..., a(11).
Among the first 10^100 terms of A132199 there are 18321 primes; of these, 3074 are distinct and 351 repeated. - Giovanni Resta, Apr 08 2016
See the crossrefs for references, links, and additional comments.

			The first duplicate in Rowland's sequence of primes A137613 = 5, 3, 11, 3, 23, 3, 47, 3, 5, ... is 3, so a(1) = 3. The second duplicate is 5, so a(2) = 5.

Giovanni Resta, Table of n, a(n) for n = 1..351

Cf. A132199, A137613, A221869.

t = {}; b1 = 7; Do[b0 = b1; b1 = b0 + GCD[n, b0]; d = b1 - b0; If[d > 1, AppendTo[t, d]], {n, 2, 10^8}]; L = {}; Do[ If[MemberQ[Take[t, n - 1], t[[n]]], AppendTo[L, t[[n]]]], {n, 2, Length[t]}]; DeleteDuplicates[L]

a(12)-a(57) from Giovanni Resta, Apr 08 2016

A231900 Omit the 1s from first differences of A084663.

Original entry on oeis.org

2, 7, 13, 5, 29, 3, 59, 3, 7, 5, 3, 131, 3, 263, 3, 17, 3, 5, 3, 19, 569, 3, 17, 3, 13, 7, 5, 3, 1181, 3, 17, 3, 2381, 3, 11, 3, 5, 3, 7, 4787, 3, 5, 3, 11, 3, 53, 3, 11, 3, 13, 19, 9689, 3, 19379, 3, 7, 5, 3, 137, 3, 13, 38921, 3, 17, 3, 7, 77867, 3, 5, 3

Offset: 1

Reinhard Zumkeller, Nov 15 2013

nonn

Terms greater than 1 in A134744.

Reinhard Zumkeller, Table of n, a(n) for n = 1..120
Eric S. Rowland, A natural prime-generating recurrence, arXiv:0710.3217 [math.NT], 2007-2008.

Cf. A137613.

a231900 n = a231900_list !! (n-1)
a231900_list = filter (> 1) a134744_list

DeleteCases[Differences[RecurrenceTable[{a[1]==8,a[n]==a[n-1]+GCD[ a[n-1],n]},a,{n,100000}]],1] (* Harvey P. Dale, Apr 12 2016 *)

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

A247090 Eric Rowland's generalization of A132199 as a rectangular array A read by upward antidiagonals.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 1, 1, 1, 1, 5, 3, 1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

L. Edson Jeffery, Nov 18 2014

Conjecture [Rowland] (paraphrased): Let A be the above array with entry A(n,k) in row n and column k. For each n, there exists an index N(n) >= 1 such that A(n,j) is either 1 or prime for all j > N(n).

			Array A begins:
1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1, ...
2, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1, ...
1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1,  1, ...
2, 3, 1, 5, 3, 1, 1, 1, 1, 11, 3,  1, ...
1, 3, 1, 5, 3, 1, 1, 1, 1, 11, 3,  1, ...
2, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3,  1, ...
1, 1, 1, 5, 3, 1, 1, 1, 1, 11, 3,  1, ...
2, 1, 1, 1, 1, 7, 1, 1, 1,  1, 1, 13, ...
1, 1, 1, 1, 1, 7, 1, 1, 1,  1, 1, 13, ...
2, 3, 1, 1, 1, 1, 1, 1, 1, 11, 3,  1, ...
1, 3, 1, 1, 1, 1, 1, 1, 1, 11, 3,  1, ...
2, 1, 1, 1, 1, 1, 1, 1, 1, 11, 3,  1, ...
...

Eric S. Rowland, A natural prime-generating recurrence, J. Integer Seq., 11 (2008), Article 08.2.8.

Cf. A106108, A132199, A137613.

(* Array A: *)
max := 13; b[n_, 1] := n; b[n_, k_] := b[n, k] = b[n, k - 1] + GCD[k, b[n, k - 1]]; Grid[Transpose[Differences[Transpose[Table[b[n, k], {n, max}, {k, max}]]]]]
(* Array antidiagonals flattened: *)
max := 13; b[n_, 1] := n; b[n_, k_] := b[n, k] = b[n, k - 1] + GCD[k, b[n, k - 1]]; Flatten[Table[Transpose[Differences[Transpose[Table[b[n, k], {n, max}, {k, max}]]]][[n - k + 1]][[k]], {n, max - 1}, {k, n}]]

cp's OEIS Frontend

A226781 Number of 1's in A132199 preceding the n-th Rowland prime, A137613(n).

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A106108 Rowland's prime-generating sequence: a(1) = 7; for n > 1, a(n) = a(n-1) + gcd(n, a(n-1)).

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

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A190894 Auxiliary c(n) sequence used to prove some properties about Rowland's sequence. c(n) has the following recursive definition: c(1) = 5, c_(n+1) = c(n) + lfp(c(n)) - 1, where lpf(.) denotes the lowest prime factor of a number.

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A221869 New primes found by Rowland's recurrence in the order of their appearance.

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A191304 Records in A132199.

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A225487 Duplicate primes found by Rowland's recurrence in the order of their reappearance.

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