A003312 -id:A003312 - cp's OEIS frontend

A061419 a(n) = ceiling(a(n-1)*3/2) with a(1) = 1.

1, 2, 3, 5, 8, 12, 18, 27, 41, 62, 93, 140, 210, 315, 473, 710, 1065, 1598, 2397, 3596, 5394, 8091, 12137, 18206, 27309, 40964, 61446, 92169, 138254, 207381, 311072, 466608, 699912, 1049868, 1574802, 2362203, 3543305, 5314958, 7972437, 11958656

Offset: 1

Henry Bottomley, May 02 2001

nonn

It appears that this sequence is the (L)-sieve transform of {3,6,9,12,...,3n,...} = A008585. (See A152009 for the definition of the (L)-sieve transform.) - John W. Layman, Jan 06 2009

			a(6) = ceiling(8*3/2) = 12.

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.30.1, p. 196.

Harry J. Smith, Table of n, a(n) for n = 1..500
Zakir Deniz, Topology of acyclic complexes of tournaments and coloring, Applicable Algebra in Engineering, Communication, March 2015, Volume 26, Issue 1-2, pp. 213-226.
A. Dubickas, On integer sequences generated by linear maps, Glasg. Math. J. 51(2) (2009), 243-252.
Don Knuth, Ambidextrous Numbers, Preprint, September 2022.
A. M. Odlyzko and H. S. Wilf, Functional iteration and the Josephus problem, Glasgow Math. J. 33(2) (1991), 235-240.
Eric Weisstein's World of Mathematics, Power Ceilings.
Index entries for sequences related to the Josephus Problem

Cf. A002379, A003312, A034082, A061418, A061420, A070885, A083286.

First differences are in A073941.

[ n eq 1 select 1 else Ceiling(Self(n-1)*3/2): n in [1..40] ]; // Klaus Brockhaus, Nov 14 2008

a:=proc(n) option remember: if n=1 then 1 else ceil(procname(n-1)*3/2) fi; end; seq(a(n),n=1..40); # Muniru A Asiru, Jun 07 2018

a=1;a=Table[a=Ceiling[a*3/2],{n,0,4!}] (* Vladimir Joseph Stephan Orlovsky, Apr 13 2010 *)
NestList[Ceiling[3#/2]&,1,39] (* Stefano Spezia, Dec 08 2024 *)

{ a=2/3; for (n=1, 500, write("b061419.txt", n, " ", a=ceil(a*3/2)) ) } \\ Harry J. Smith, Jul 22 2009

from itertools import islice
def A061419_gen(): # generator of terms
    a = 2
    while True:
        yield a-1
        a += a>>1
A061419_list = list(islice(A061419_gen(),70)) # Chai Wah Wu, Sep 20 2022

a(n) = A061418(n) - 1 = floor(K*(3/2)^n) where K = 1.08151366859...
The constant K is (2/3)*K(3) (see A083286). - Ralf Stephan, May 29 2003
a(1) = 1, a(n) = A070885(n)/3. - Benoit Cloitre, Aug 18 2002
a(n) = ceiling((a(n-1) + a(n-2))*9/10) - Franklin T. Adams-Watters, May 01 2006

A061418 a(n) = floor(a(n-1)*3/2) with a(1) = 2.

Original entry on oeis.org

2, 3, 4, 6, 9, 13, 19, 28, 42, 63, 94, 141, 211, 316, 474, 711, 1066, 1599, 2398, 3597, 5395, 8092, 12138, 18207, 27310, 40965, 61447, 92170, 138255, 207382, 311073, 466609, 699913, 1049869, 1574803, 2362204, 3543306, 5314959, 7972438

Offset: 1

Henry Bottomley, May 02 2001

Can be stated as the number of animals starting from a single pair if any pair of animals can produce a single offspring (as in the game Minecraft, if the player allows offspring to fully grow before breeding again). - Denis Moskowitz, Dec 05 2012

Maximum number of partial products that can be added in a Wallace tree multiplier with n-1 full adder stages. - Chinmaya Dash, Aug 19 2020

			a(6) = floor(9*3/2) = 13.

Iain Fox, Table of n, a(n) for n = 1..1000 (first 500 terms from Harry J. Smith)
Don Knuth, Ambidextrous Numbers, Preprint, September 2022.
M. van de Vel, Determination of msd(L^n), J. Algebraic Combin. 9(2) (1999), 161-171. See Table 5. - _N. J. A. Sloane_, Mar 26 2012
Mark van Wijk, The Quest for the Best Thread-Safe Java List, Univ. of Twente (Netherlands 2022).
Wikipedia, Wallace tree.

Cf. A002379, A003312, A034082, A061419, A083286.

First differences are in A073941.

[ n eq 1 select 2 else Floor(Self(n-1)*(3/2)): n in [1..39] ]; // Klaus Brockhaus, Nov 14 2008

{ a=4/3; for (n=1, 500, a=a*3\2; write("b061418.txt", n, " ", a) ) } \\ Harry J. Smith, Jul 22 2009

first(n) = my(v=vector(n)); v[1]=2; for(i=2, n, v[i]=v[i-1]*3\2); v \\ Iain Fox, Jul 15 2022

from itertools import islice
def A061418_gen(): # generator of terms
    a = 2
    while True:
        yield a
        a += a>>1
A061418_list = list(islice(A061418_gen(),70)) # Chai Wah Wu, Sep 20 2022

a(n) = A061419(n) + 1 = ceiling(K*(3/2)^n) where K = 1.08151366859...

The constant K is (2/3)*K(3) (see A083286). - Ralf Stephan, May 29 2003

A006999 Partitioning integers to avoid arithmetic progressions of length 3.

Original entry on oeis.org

0, 1, 2, 4, 7, 11, 17, 26, 40, 61, 92, 139, 209, 314, 472, 709, 1064, 1597, 2396, 3595, 5393, 8090, 12136, 18205, 27308, 40963, 61445, 92168, 138253, 207380, 311071, 466607, 699911, 1049867, 1574801, 2362202, 3543304, 5314957, 7972436

Offset: 0

N. J. A. Sloane, D. R. Hofstadter, and James Propp, Jul 15 1977

a(n) = A006997(3^n-1).
It appears that, aside from the first term, this is the (L)-sieve transform of A016789 ={2,5,8,11,...,3n+2....}. This has been verified up to a(30)=311071. See A152009 for the definition of the (L)-sieve transform. - John W. Layman, Nov 20 2008
a(n) is also the largest-index square reachable in n jumps if we start at square 0 of the Infinite Sidewalk. - Jose Villegas, Mar 27 2023

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
B. Chen, R. Chen, J. Guo, S. Lee et al, On Base 3/2 and its sequences, arXiv:1808.04304 [math.NT], 2018.
Joseph Gerver, James Propp and Jamie Simpson, Greedily partitioning the natural numbers into sets free of arithmetic progressions Proc. Amer. Math. Soc. 102 (1988), no. 3, 765-772.
D. R. Hofstadter, Eta-Lore [Cached copy, with permission]
D. R. Hofstadter, Pi-Mu Sequences [Cached copy, with permission]
D. R. Hofstadter and N. J. A. Sloane, Correspondence, 1977 and 1991
James Propp and N. J. A. Sloane, Email, March 1994
Jane Street, Traversing the Infinite Sidewalk (2023).

Cf. A061419, A061418, A005428 (first differences), A083286.
Cf. A006997, A016789, A152009.
Cf. A003312.

a006999 n = a006999_list !! n
a006999_list = 0 : map ((`div` 2) . (+ 2) . (* 3)) a006999_list
-- Reinhard Zumkeller, Oct 26 2011

a[0] = 0; a[n_] := a[n] = Floor[(3 a[n-1] + 2)/2];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Aug 01 2018 *)

PARI
```
a(n)=if(n<1,0,floor((3*a(n-1)+2)/2))
```

a(n) = A061419(n) - 1.
a(n) = A061418(n) - 2.
a(n) = floor((3a(n-1)+2)/2).
a(n) = -1 + floor(c*(3/2)^n) where c=1.0815136... - Benoit Cloitre, Jan 10 2002; this constant c is 2/3*K(3) (see A083286). - Ralf Stephan, May 29 2003
a(n+1) = (3*a(n))/2+1 if a(n) is even. a(n+1) = (3*a(n)+1)/2 if a(n) is odd. - Miquel Cerda, Jun 15 2019

More terms from James Sellers, Feb 06 2000

A070885 a(n) = (3/2)a(n-1) if a(n-1) is even; (3/2)(a(n-1)+1) if a(n-1) is odd.

Original entry on oeis.org

1, 3, 6, 9, 15, 24, 36, 54, 81, 123, 186, 279, 420, 630, 945, 1419, 2130, 3195, 4794, 7191, 10788, 16182, 24273, 36411, 54618, 81927, 122892, 184338, 276507, 414762, 622143, 933216, 1399824, 2099736, 3149604, 4724406, 7086609, 10629915

Offset: 1

Eric W. Weisstein, May 14 2002

nonn

The smallest positive number such that A024629(a(n)) has n digits, per page 9 of the Tanton reference in Links. - Glen Whitney, Sep 17 2017

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, 2002, p. 123.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
B. Chen, R. Chen, J. Guo, S. Lee et al., On Base 3/2 and its Sequences, arXiv:1808.04304 [math.NT], 2018.
James Tanton, Exploding Dots, Chapter 9
Eric Weisstein's World of Mathematics, Wolfram Sequences

The constant K is 2/3*K(3) (see A083286). - Ralf Stephan, May 29 2003
Cf. A003312.
Cf. A081848.
Cf. A205083 (parity of terms).

a070885 n = a070885_list !! (n-1)
a070885_list = 1 : map (flip (*) 3 . flip div 2 . (+ 1)) a070885_list
-- Reinhard Zumkeller, Sep 05 2014

A070885 := proc(n)
    option remember;
    if n = 1 then
        return 1;
    elif type(procname(n-1),'even') then
        procname(n-1) ;
    else
        procname(n-1)+1 ;
    end if;
    %*3/2 ;
end proc:
seq(A070885(n),n=1..80) ; # R. J. Mathar, Jun 18 2018

NestList[If[EvenQ[#],3/2 #,3/2 (#+1)]&,1,40] (* Harvey P. Dale, May 18 2018 *)

from itertools import islice
def A070885_gen(): # generator of terms
    a = 1
    while True:
        yield a
        a += (a+1>>1)+(a&1)
A070885_list = list(islice(A070885_gen(),70)) # Chai Wah Wu, Sep 20 2022

For n > 1, a(n) = 3*A061419(n) = 3*floor(K*(3/2)^n) where K=1.08151366859... - Benoit Cloitre, Aug 18 2002
a(n) = 3*ceiling(a(n-1)/2). - Benoit Cloitre, Apr 25 2003
a(n+1) = a(n) + A081848(n), for n > 1. - Reinhard Zumkeller, Sep 05 2014

A100585 a(n+1) = a(n)+floor(a(n)/3), a(1) = 3.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 13, 17, 22, 29, 38, 50, 66, 88, 117, 156, 208, 277, 369, 492, 656, 874, 1165, 1553, 2070, 2760, 3680, 4906, 6541, 8721, 11628, 15504, 20672, 27562, 36749, 48998, 65330, 87106, 116141, 154854, 206472, 275296, 367061, 489414, 652552

Offset: 1

N. J. A. Sloane, Dec 01 2004

nonn

Original definition: Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off every 4th term. Repeat, always crossing off every 4th term of those that remain. The numbers that are left form the sequence.

Can be stated as the number of animals starting from a single trio if any trio of animals can produce a single offspring. See A061418 for the equivalent sequence for pairs of animals. - Luca Khan, Sep 05 2024

Robert Israel, Table of n, a(n) for n = 1..7987
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #6 with K=4. [Annotated and scanned copy]
Index entries for sequences generated by sieves

Cf. A003309, A003310, A100464, A100562, A003312, A003311, A052548, A100586, A061418.

R:= 3: x:= 3:
for i from 2 to 100 do x:= x + floor(x/3); R:= R,x od:
R; # Robert Israel, Sep 09 2024

t = Range[3, 2500000]; r = {}; While[Length[t] > 0, AppendTo[r, First[t]]; t = Drop[t, {1, -1, 4}];]; r (* Ray Chandler, Dec 02 2004 *)
NestList[#+Floor[#/3]&,3,50] (* Harvey P. Dale, Jan 14 2019 *)

a(n,s=3)=for(i=2,n,s+=s\3);s \\ M. F. Hasler, Oct 06 2014

a(1)=3, a(n+1) = a(n) + floor(a(n)/3). - Ben Paul Thurston, Jan 09 2008

More terms from Ray Chandler, Dec 02 2004

Simpler definition from M. F. Hasler, Oct 06 2014

A003311 Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off the first, (M+1)st, (2M+1)st, (3M+1)st, etc. Repeat. The numbers that are left form the sequence.

Original entry on oeis.org

3, 5, 8, 11, 15, 18, 23, 27, 32, 38, 42, 47, 53, 57, 63, 71, 75, 78, 90, 93, 98, 105, 113, 117, 123, 132, 137, 140, 147, 161, 165, 168, 176, 183, 188, 197, 206, 212, 215, 227, 233, 237, 243, 252, 258, 267, 278, 282, 287, 293, 303, 312, 317, 323

Offset: 1

N. J. A. Sloane

nonn

			The first few sieving stages are as follows:
  3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
  3 X 5 6 X 8 9 XX 11 12 XX 14 15 XX 17 18 XX 20 ...
  3 X 5 X X 8 9 XX 11 12 XX XX 15 XX 17 18 XX 20 ...
  3 X 5 X X 8 X XX 11 12 XX XX 15 XX 17 18 XX 20 ...
  3 X 5 X X 8 X XX 11 XX XX XX 15 XX 17 18 XX 20 ...
  3 X 5 X X 8 X XX 11 XX XX XX 15 XX XX 18 XX 20 ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. Based on a misreading of Sieve #3. A100464 is the correct version. [Annotated and scanned copy]
Index entries for sequences generated by sieves

Cf. A003309, A003310, A100464, A003312, A100562, A100585, A052548.

a003311 n = a003311_list !! (n-1)
a003311_list = f [3..] where
   f (x:xs) = x : f (g xs) where
     g zs = us ++ g vs where (_:us, vs) = splitAt x zs
-- Reinhard Zumkeller, Nov 12 2014

Entry revised Nov 29 2004

A100586 Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off every 5th term. Repeat, always crossing off every 5th term of those that remain. The numbers that are left form the sequence.

Original entry on oeis.org

3, 4, 5, 6, 7, 9, 11, 14, 17, 21, 26, 32, 40, 50, 62, 77, 96, 120, 150, 187, 234, 292, 365, 456, 570, 712, 890, 1112, 1390, 1737, 2171, 2714, 3392, 4240, 5300, 6625, 8281, 10351, 12939, 16174, 20217, 25271, 31589, 39486, 49357, 61696, 77120

Offset: 1

N. J. A. Sloane, Dec 01 2004

nonn

Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #6 with K=5. [Annotated and scanned copy]
Index entries for sequences generated by sieves

Cf. A003309, A003310, A100464, A100562, A003312, A003311, A052548, A100585.

t = Range[3, 80000]; r = {}; While[Length[t] > 0, AppendTo[r, First[t]]; t = Drop[t, {1, -1, 5}];]; r (* Ray Chandler, Dec 02 2004 *)

cp's OEIS Frontend

A061419 a(n) = ceiling(a(n-1)*3/2) with a(1) = 1.

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A061418 a(n) = floor(a(n-1)*3/2) with a(1) = 2.

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A006999 Partitioning integers to avoid arithmetic progressions of length 3.

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A070885 a(n) = (3/2)*a(n-1) if a(n-1) is even; (3/2)*(a(n-1)+1) if a(n-1) is odd.

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A100585 a(n+1) = a(n)+floor(a(n)/3), a(1) = 3.

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Maple

Mathematica

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A003311 Write down the numbers from 3 to infinity. Take next number, M say, that has not been crossed off. Counting through the numbers that have not yet been crossed off after that M, cross off the first, (M+1)st, (2M+1)st, (3M+1)st, etc. Repeat. The numbers that are left form the sequence.

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A070885 a(n) = (3/2)a(n-1) if a(n-1) is even; (3/2)(a(n-1)+1) if a(n-1) is odd.