A106108 -id:A106108 - cp's OEIS frontend

A166945 Records of first differences of A166944.

2, 3, 7, 13, 43, 139, 313, 661, 1321, 2659, 5419, 10891, 22039, 44383, 88801, 177841, 355723, 713833, 1427749, 2860771, 5725453, 11461141, 22933441, 45895573, 91793059, 183616423, 367232911, 734482123, 1468965061, 2937930211, 5875882249, 11751795061, 23503590559, 47007181621, 94014363763

Offset: 1

Vladimir Shevelev, Oct 24 2009, Nov 05 2009

nonn

Conjecture. Each term of the sequence is the greater of a pair of twin primes (A006512).

E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol.11(2008), Article 08.2.8. arXiv:0710.3217 [math.NT]
V. Shevelev, An infinite set of generators of primes based on the Rowland idea and conjectures concerning twin primes, arXiv:0910.4676 [math.NT], 2009.
V. Shevelev, Three theorems on twin primes, arXiv:0911.5478 [math.NT], 2009-2010.

Cf. A166944, A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293.

Reap[Print[old = r = 2]; Sow[old]; For[n = 2, n <= 10^6, n++, d = GCD[old, If[OddQ[n], n-2, n]]; If[d>r, r=d; Print[d]; Sow[d]]; old += d]][[2, 1]] (* Jean-François Alcover, Nov 03 2018, from PARI *)

print1(old=r=2); for(n=2,1e11, d=gcd(old,if(n%2,n-2,n)); if(d>r, r=d; print1(", "d)); old+=d) \\ Charles R Greathouse IV, Oct 13 2017

6 more terms from R. J. Mathar, Nov 19 2009; extension beginning with a(19) from Benoit Cloitre (private communication to Vladimir Shevelev)
a(25), a(26) from D. S. McNeil, Dec 13 2010
a(27)-a(30) from Charles R Greathouse IV, Oct 13 2017
a(31)-a(35) from Charles R Greathouse IV, Oct 17 2017

A116533 a(1)=1, a(2)=2, for n > 2 if a(n-1) is prime, then a(n) = 2*a(n-1), otherwise a(n) = a(n-1) - 1.

Original entry on oeis.org

1, 2, 4, 3, 6, 5, 10, 9, 8, 7, 14, 13, 26, 25, 24, 23, 46, 45, 44, 43, 86, 85, 84, 83, 166, 165, 164, 163, 326, 325, 324, 323, 322, 321, 320, 319, 318, 317, 634, 633, 632, 631, 1262, 1261, 1260, 1259, 2518, 2517, 2516, 2515, 2514, 2513, 2512, 2511, 2510, 2509, 2508

Offset: 1

Rodolfo Kurchan, Mar 26 2006

nonn

For n >= 3, using Wilson's theorem, a(n) = a(n-1) + (-1)^r*gcd(a(n-1), W), where W = A038507(a(n-1) - 1), and r=1 if gcd(a(n-1), W) = 1 and r=0 otherwise. - Vladimir Shevelev, Aug 07 2009

Cf. A006992, A055496, A080359, A104272, A106108, A132199. - Vladimir Shevelev, Aug 07 2009

a[1]:=1: a[2]:=2: for n from 3 to 60 do if isprime(a[n-1])=true then a[n]:=2*a[n-1] else a[n]:=a[n-1]-1 fi od: seq(a[n],n=1..60); # Emeric Deutsch, Apr 02 2006

More terms from Emeric Deutsch, Apr 02 2006

A163961 First differences of A116533.

Original entry on oeis.org

1, 2, -1, 3, -1, 5, -1, -1, -1, 7, -1, 13, -1, -1, -1, 23, -1, -1, -1, 43, -1, -1, -1, 83, -1, -1, -1, 163, -1, -1, -1, -1, -1, -1, -1, -1, -1, 317, -1, -1, -1, 631, -1, -1, -1, 1259, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 2503, -1, -1, -1, 5003, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1

Offset: 1

Vladimir Shevelev, Aug 07 2009, Aug 14 2009

sign

Ignoring the +-1 terms, we obtain the sequence of Bertrand's primes A006992. If we consider sequences A_i={a_i(n)}, i=1,2,... with the same constructions as A116533, but with initials a_1(1)=2, a_2(1)=11, a_3(1)=17,..., a_m(1)=A164368(m),..., then the union of A_1,A_2,... contains all primes.

Cf. A116533, A006992, A055496, A080359, A104272, A106108, A132199, A164368

A116533 := proc(n) option remember; if n <=2 then n; else if isprime(procname(n-1)) then 2*procname(n-1) ; else procname(n-1)-1 ; end if; end if; end proc:
A163961 := proc(n) A116533(n+1)-A116533(n) ; end proc: # R. J. Mathar, Sep 03 2011

Differences@ Prepend[NestList[If[PrimeQ@ #, 2 #, # - 1] &, 2, 90], 1] (* Michael De Vlieger, Dec 06 2018 *)

a116533(n) = if(n==1, 1, if(n==2, 2, if(ispseudoprime(a116533(n-1)), 2*a116533(n-1), a116533(n-1)-1)))
a(n) = a116533(n+1)-a116533(n) \\ Felix Fröhlich, Dec 06 2018

lista(nn) = {va = vector(nn); va[1] = 1; va[2] = 2; for (n=3, nn, va[n] = if (isprime(va[n-1]), 2*va[n-1], va[n-1]-1);); vector(nn-1, n, va[n+1] - va[n]);} \\ Michel Marcus, Dec 07 2018

A163963 First differences of A080735.

Original entry on oeis.org

1, 2, 1, 5, 1, 11, 1, 23, 1, 47, 1, 1, 1, 97, 1, 1, 1, 197, 1, 1, 1, 397, 1, 1, 1, 797, 1, 1, 1, 1597, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3203, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 6421, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 12853, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25717, 1, 1, 1, 51437, 1, 1, 1

Offset: 1

Vladimir Shevelev, Aug 07 2009

nonn

Ignoring the 1 terms we obtain A055496. If we consider sequences A_i={a_i(n)}, i=1,2,... with the same constructions as A080735, but with initials a_1(1)=2, a_2(1)=3, a_3(1)=13,..., a_m(1)=A080359(m),..., then the union of A_1,A_2,... contains all primes.

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A116533, A163961, A080735, A006992, A055496, A080359, A104272, A106108, A132199.

A080735 := proc(n) option remember; local p ; if n = 1 then 1; else p := procname(n-1) ; if isprime(p) then 2*p; else p+1 ; end if; end if; end proc: A163963 := proc(n) A080735(n+1)-A080735(n) ; end: seq(A163963(n),n=1..100) ; # R. J. Mathar, Nov 05 2009

Differences@ NestList[If[PrimeQ@ #, 2 #, # + 1] &, 1, 87] (* Michael De Vlieger, Dec 06 2018, after Harvey P. Dale at A080735 *)

lista(nn) = {my(va = vector(nn)); va[1] = 1; for (n=2, nn, va[n] = if (isprime(va[n-1]), 2*va[n-1], va[n-1]+1);); vector(nn-1, n, va[n+1] - va[n]);} \\ Michel Marcus, Dec 06 2018

More terms from R. J. Mathar, Nov 05 2009

A135508 a(n) = x(n+1)/x(n) - 2 where x(1)=1 and x(n) = 2*x(n-1) + lcm(x(n-1),n).

Original entry on oeis.org

2, 3, 1, 1, 1, 7, 2, 1, 1, 11, 1, 1, 7, 1, 1, 17, 1, 1, 1, 7, 11, 23, 1, 1, 1, 1, 7, 29, 1, 1, 2, 11, 17, 7, 1, 37, 1, 1, 1, 41, 7, 1, 11, 1, 23, 47, 1, 1, 1, 17, 1, 53, 1, 1, 1, 1, 29, 59, 1, 1, 1, 1, 1, 1, 1, 67, 17, 1, 1, 71, 1, 1, 37, 1, 1, 1, 1, 79, 1, 1, 41, 83, 1, 1, 1, 29, 1, 89, 1, 1, 1, 1

Offset: 1

Benoit Cloitre, Feb 09 2008

nonn

This sequence has properties related to primes and especially to twin primes. For instance sequence consists of 1's or primes only. 2 occurs infinitely many times, largest primes in twin pairs never occur, other primes occur finitely many times...

For each prime p that appears in the sequence, its first appearance is at a(p-1). - Bill McEachen, Sep 04 2022

Markus Schepke, Über Primzahlerzeugende Folgen, Thesis, U. Hannover, 2009.

Cf. A106108.

f[1] := 1; f[n_] := 2*f[n - 1] + LCM[f[n - 1], n]; Table[f[n + 1]/f[n] - 2, {n, 1, 10}] (* G. C. Greubel, Oct 16 2016 *)

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

a(2*4^k) = 2, k >= 0.

A166944 a(1)=2; a(n) = a(n-1) + gcd(n, a(n-1)) if n is even, a(n) = a(n-1) + gcd(n-2, a(n-1)) if n is odd.

Original entry on oeis.org

2, 4, 5, 6, 9, 12, 13, 14, 21, 22, 23, 24, 25, 26, 39, 40, 45, 54, 55, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 129, 130, 135, 138, 139, 140, 147, 148, 149, 150, 151, 152, 153, 154, 155, 160, 161, 162, 163

Offset: 1

Vladimir Shevelev, Oct 24 2009

nonn

Conjecture: Every record of differences a(n)-a(n-1) more than 5 is the greater of twin primes (A006512).

Harvey P. Dale, Table of n, a(n) for n = 1..1000
E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol.11(2008), Article 08.2.8. arXiv:0710.3217 [math.NT]
V. Shevelev, An infinite set of generators of primes based on the Rowland idea and conjectures concerning twin primes, arXiv:0910.4676 [math.NT], 2009. - _Vladimir Shevelev_, Oct 27 2009
V. Shevelev, Three theorems on twin primes, arXiv:0911.5478 [math.NT], 2009-2010. - _Vladimir Shevelev_, Dec 03 2009

Cf. A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293.

A166944 := proc(n) option remember; if n = 1 then 2; else p := procname(n-1) ; if type(n,'even') then p+igcd(n,p) ; else p+igcd(n-2,p) ; end if; end if; end proc: # R. J. Mathar, Sep 03 2011

nxt[{n_,a_}]:={n+1,If[OddQ[n],a+GCD[n+1,a],a+GCD[n-1,a]]}; Transpose[ NestList[ nxt,{1,2},70]][[2]] (* Harvey P. Dale, Feb 10 2015 *)

print1(a=2); for(n=2, 100, d=gcd(a, if(n%2, n-2, n)); print1(", "a+=d)) \\ Charles R Greathouse IV, Oct 13 2017

Terms beginning with a(18) corrected by Vladimir Shevelev, Nov 10 2009

A135504 a(1)=1; for n>1, a(n) = a(n-1) + lcm(a(n-1),n).

Original entry on oeis.org

1, 3, 6, 18, 108, 216, 1728, 3456, 6912, 41472, 497664, 995328, 13934592, 27869184, 167215104, 334430208, 6019743744, 12039487488, 240789749760, 481579499520, 963158999040, 11557907988480, 277389791723520, 554779583447040

Offset: 1

Benoit Cloitre, Feb 09 2008, Feb 10 2008

This sequence has properties related to primes. For instance: a(n+1)/a(n)-1 consists of 1's or primes only. Any prime p>=3 divides a(n) for the first time when n=p*w(p)-1 where w(p) is the least positive integer such that p*w(p)-1 is prime.
See A135506 for more comments and references.
Partial sums of A074179. - David Radcliffe, Jun 23 2025

T. D. Noe, Table of n, a(n) for n=1..100

Cf. A074179, A135506, A361460, A361461, A361463, A361464, A361470.

Cf. also A106108.

a135504 n = a135504_list !! (n-1)
a135504_list = 1 : zipWith (+)
                   a135504_list (zipWith lcm a135504_list [2..])
-- Reinhard Zumkeller, Oct 03 2012

a[1] = 1; a[n_] := a[n] = a[n-1] + LCM[a[n-1], n]; Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Dec 16 2011 *)
RecurrenceTable[{a[1]==1,a[n]==a[n-1]+LCM[a[n-1],n]},a,{n,30}] (* Harvey P. Dale, Mar 03 2013 *)

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

from sympy import lcm
l=[0, 1]
for n in range(2, 101):
    x=l[n - 1]
    l.append(x + lcm(x, n))
print(l) # Indranil Ghosh, Jun 27 2017

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

Original entry on oeis.org

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

Jonathan Sondow, Jan 29 2008, Jan 30 2008

nonn

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

			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)

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.

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.

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

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

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 *)

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

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

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

A167053 a(1)=3; for n > 1, a(n) = 1 + a(n-1) + gcd( a(n-1)*(a(n-1)+2), A073829(a(n-1)) ).

Original entry on oeis.org

3, 19, 39, 81, 165, 333, 335, 673, 1347, 1349, 1351, 1353, 1355, 1357, 1359, 2721, 2723, 2725, 2727, 5457, 5459, 5461, 5463, 5465, 5467, 5469, 10941, 10943, 10945, 10947, 21897, 21899, 21901, 21903, 21905, 21907, 21909, 43821, 43823, 43825, 43827, 43829, 43831

Offset: 1

Vladimir Shevelev, Oct 27 2009

nonn

The first differences are 16, 20, 42, etc. They are either 2 or in A075369 or in A008864, see A167054.

A proof follows from Clement's criterion of twin primes.

			a(2) = 1 + 3 + gcd(3*5, 4*(2! + 1) + 3) = 19.

E. Trost, Primzahlen, Birkhäuser-Verlag, 1953, pages 30-31.

Amiram Eldar, Table of n, a(n) for n = 1..206
P. A. Clement, Congruences for sets of primes, Amer. Math. Monthly, 56 (1949), 23-25.

Cf. A073829, A008864, A167054.

Cf. A166944, A166945, A116533, A163961, A163963, A084662, A084663, A106108, A132199, A134162, A135506, A135508, A118679, A120293.

A073829 := proc(n) n+4*((n-1)!+1) ; end proc:
A167053 := proc(n) option remember ; local aprev; if n = 1 then 3; else aprev := procname(n-1) ; 1+aprev+gcd(aprev*(aprev+2),A073829(aprev)) ; end if; end proc:
seq(A167053(n),n=1..60) ; # R. J. Mathar, Dec 17 2009

A073829[n_] := 4((n-1)! + 1) + n;
a[1] = 3;
a[n_] := a[n] = 1 + a[n-1] + GCD[a[n-1] (a[n-1] + 2), A073829[a[n-1]]];
Array[a, 60] (* Jean-François Alcover, Mar 25 2020 *)

Definition shortened and values from a(4) on replaced by R. J. Mathar, Dec 17 2009

A167168 Sequence of prime gaps which characterize Rowland sequences of prime-generating recurrences.

Original entry on oeis.org

3, 7, 17, 19, 31, 43, 53, 67, 71, 79, 97, 103, 109, 113, 127, 137, 151, 163, 173, 181, 191, 197, 199, 211, 229, 239, 241, 251, 257, 269, 271, 283, 293, 317, 331, 337, 349, 367, 373

Offset: 1

Vladimir Shevelev, Oct 29 2009

nonn

Consider the Rowland sequences with recurrence N(n)= N(n-1)+gcd(n,N(n-1)).
For some of these, like the prototypical A106108, the first differences N(n)-N(n-1) are always 1 or primes.
If for some position p (a prime) N(p-1)=2*p, then the arXiv preprint shows that N is indeed in that class of prime-generating sequences.
Since then N(p)=N(p-1)+p, the prime p characterizes at the same time the gap (first difference) and location in the sequence.
In the same sequence at some larger value of p, we may again have N(p-1)=2*p. In these cases, we put all these p's satisfying that equation into a generator class.
For each of the generator classes, the OEIS sequence shows the smallest member (prime) in that class. So this is a trace of how many essentially different sequences with this N(p-1)=2*p property exist.

			We put a(1)=3 since the N-sequence 4, 6, 9, 10, 15, 18, 19, 20.. = A084662 (essentially the same as A106108) has a first difference of p=3 at position p-1=2, N(2)=2*3.
It has a first difference of p=5 at p-1=4, a first difference of p=11 at p=10, so we put {3,5,11,23,..} into that class. This leaves p=7=a(2) as the lowest prime to be covered by the next class. This is first realized by N = 8, 10, 11, 12, 13, 14, 21, 22, 23, 24, 25, 26, 39.. = A084663. Here N(12)=2*13, so p=13 is in the same class as p=7, namely {7,13,29,59,131,..}. This leaves p=17=a(3) to be the smallest member in a new class, namely {17,41,83,167,..}.

E. S. Rowland, A natural prime-generating recurrence, Journal of Integer Sequences, Vol. 11 (2008), Article 08.2.8
V. Shevelev, A new generator of primes based on the Rowland idea, arXiv:0910.4676 [math.NT], 2009.

Cf. A106108, A167053, A116533, A163963, A084662, A084663, A134162.

Edited, a(1) set to 3, 37 replaced by 31, and extended beyond 53 by R. J. Mathar, Dec 17 2009

cp's OEIS Frontend

A166945 Records of first differences of A166944.

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A116533 a(1)=1, a(2)=2, for n > 2 if a(n-1) is prime, then a(n) = 2*a(n-1), otherwise a(n) = a(n-1) - 1.

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A163961 First differences of A116533.

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A163963 First differences of A080735.

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A135508 a(n) = x(n+1)/x(n) - 2 where x(1)=1 and x(n) = 2*x(n-1) + lcm(x(n-1),n).

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A166944 a(1)=2; a(n) = a(n-1) + gcd(n, a(n-1)) if n is even, a(n) = a(n-1) + gcd(n-2, a(n-1)) if n is odd.

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A135504 a(1)=1; for n>1, a(n) = a(n-1) + lcm(a(n-1),n).

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