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

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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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.

Examples

			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 ...
		

References

• 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).

Links

• 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

Crossrefs

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

Programs

Haskell

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

Maple

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

Mathematica

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

PARI

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

Python

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

Python

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

Extensions

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