A132199 -id:A132199 - cp's OEIS frontend

A191304 Records in A132199.

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

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}]]

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

Original entry on oeis.org

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

A166945 Records of first differences of A166944.

Original entry on oeis.org

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

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

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

cp's OEIS Frontend

A191304 Records in A132199.

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

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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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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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