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A003312 a(1) = 3; for n>0, a(n+1) = a(n) + floor((a(n)-1)/2).
(Formerly M2308)
8

%I M2308

%S 3,4,5,7,10,14,20,29,43,64,95,142,212,317,475,712,1067,1600,2399,3598,

%T 5396,8093,12139,18208,27311,40966,61448,92171,138256,207383,311074,

%U 466610,699914,1049870,1574804,2362205,3543307,5314960,7972439,11958658,17937986,26906978

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

%C 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 1000-th term, and was unsolved for several years.

%D 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.

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

%H T. D. Noe and N. J. A. Sloane, <a href="/A003312/b003312.txt">Table of n, a(n) for n = 1..1000</a> [First 500 terms from T. D. Noe]

%H Popular Computing (Calabasas, CA), <a href="/A003309/a003309.pdf">Sieves: Problem 43</a>, Vol. 2 (No. 13, Apr 1974), pp. 6-7. This is Sieve #5. [Annotated and scanned copy]

%H <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>

%e The first few sieving stages are as follows:

%e 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...

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

%e 3 4 5 X 7 X X 10 11 XX XX 14 XX 16 XX XX 19 20 ...

%e 3 4 5 X 7 X X 10 XX XX XX 14 XX 16 XX XX XX 20 ...

%e 3 4 5 X 7 X X 10 XX XX XX 14 XX XX XX XX XX 20 ...

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

%t NestList[#+Floor[(#-1)/2]&,3,50] (* _Harvey P. Dale_, Mar 18 2011 *)

%o (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

%o (Haskell)

%o a003312 n = a003312_list !! (n-1)

%o a003312_list = sieve [3..] where

%o sieve :: [Integer] -> [Integer]

%o sieve (x:xs) = x : (sieve $ xOff xs)

%o xOff :: [Integer] -> [Integer]

%o xOff (x:x':_:xs) = x : x': (xOff xs)

%o -- _Reinhard Zumkeller_, Feb 21 2011

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

%K nonn,easy,nice

%O 1,1

%A _N. J. A. Sloane_

%E Entry revised Dec 01 2004 and May 10 2015.

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Last modified July 30 18:15 EDT 2015. Contains 260130 sequences.