A100562 -id:A100562 - cp's OEIS frontend

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

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

A003312 a(1) = 3; for n>0, a(n+1) = a(n) + floor((a(n)-1)/2).

Original entry on oeis.org

3, 4, 5, 7, 10, 14, 20, 29, 43, 64, 95, 142, 212, 317, 475, 712, 1067, 1600, 2399, 3598, 5396, 8093, 12139, 18208, 27311, 40966, 61448, 92171, 138256, 207383, 311074, 466610, 699914, 1049870, 1574804, 2362205, 3543307, 5314960, 7972439, 11958658, 17937986, 26906978

Offset: 1

N. J. A. Sloane

This sequence was originally defined in Popular Computing in 1974 by a sieve, as follows. 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 third term. Repeat, always crossing off every third term of those that remain. The numbers that are left form the sequence. The recurrence given here for the sequence was found by Colin Mallows. The problem asked for the 1000th term, and was unsolved for several years.

			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 4 5 X 7 8 X 10 11 XX 13 14 XX 16 17 XX 19 20 ...
  3 4 5 X 7 X X 10 11 XX XX 14 XX 16 XX XX 19 20 ...
  3 4 5 X 7 X X 10 XX XX XX 14 XX 16 XX XX XX 20 ...
  3 4 5 X 7 X X 10 XX XX XX 14 XX XX XX XX XX 20 ...

Popular Computing (Calabasas, CA), Problem 43, Sieves, sieve #5, Vol. 2 (No. 13, Apr 1974), pp. 6-7; Vol. 2 (No. 17, Aug 1974), page 16; Vol. 5 (No. 51, Jun 1977), p. 17.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

N. J. A. Sloane, Table of n, a(n) for n = 1..1000 (first 500 terms from T. D. Noe)
Popular Computing (Calabasas, CA), Sieves: Problem 43, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #5. [Annotated and scanned copy]
H. P. Robinson, C. L. Mallows, & N. J. A. SloaneCorrespondence, 1975
Index entries for sequences generated by sieves

Cf. A003309, A003310, A100464, A100562, A006999, A061418, A070885, A003311.

a003312 n = a003312_list !! (n-1)
a003312_list = sieve [3..] where
   sieve :: [Integer] -> [Integer]
   sieve (x:xs) = x : (sieve $ xOff xs)
   xOff :: [Integer] -> [Integer]
   xOff (x:x':_:xs) = x : x': (xOff xs)
-- Reinhard Zumkeller, Feb 21 2011

f:=proc(n) option remember; if n=1 then RETURN(3) fi; f(n-1)+floor( (f(n-1)-1)/2 ); end;

NestList[#+Floor[(#-1)/2]&,3,50]  (* Harvey P. Dale, Mar 18 2011 *)

v=vector(100); v[1]=3; for(n=2, #v, v[n]=floor((3*v[n-1]-1)/2)); v \\ Clark Kimberling, Dec 30 2010

l=[0, 3]
for n in range(2, 101):
    l.append(l[n - 1] + (l[n - 1] - 1)//2)
print(l[1:]) # Indranil Ghosh, Jun 09 2017

from itertools import islice
def A003312_gen(): # generator of terms
    a = 3
    while True:
        yield a
        a += a-1>>1
A003312_list = list(islice(A003312_gen(),30)) # Chai Wah Wu, Sep 21 2022

Entry revised by N. J. A. Sloane, Dec 01 2004 and May 10 2015

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

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

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A003312 a(1) = 3; for n>0, a(n+1) = a(n) + floor((a(n)-1)/2).

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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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