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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 originally defined in the 1974 reference 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 was found by _Colin Mallows_.

%D "Sieves", Popular Computing (Calabasas, CA), Vol. 2 (No. 13, Apr 1974), pp. 6-7; sieve #5.

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

%D Solution to Problem 170, Popular Computing (Calabasas, CA), Vol. 5 (No. 51, Jun 1977), pp. 17.

%H T. D. Noe, <a href="/A003312/b003312.txt">Table of n, a(n) for n=1..500</a>

%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

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Last modified July 22 22:03 EDT 2014. Contains 244844 sequences.