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 A003312 a(1) = 3; for n>0, a(n+1) = a(n) + floor((a(n)-1)/2). (Formerly M2308) 8
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 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. 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 T. D. Noe and 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 EXAMPLE 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 ... 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 *) PROG (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 (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 (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 CROSSREFS Cf. A003309, A003310, A100464, A100562, A006999, A061418, A070885, A003311. Sequence in context: A073957 A309916 A162311 * A022440 A347805 A088130 Adjacent sequences: A003309 A003310 A003311 * A003313 A003314 A003315 KEYWORD nonn,easy,nice AUTHOR N. J. A. Sloane EXTENSIONS Entry revised by N. J. A. Sloane, Dec 01 2004 and May 10 2015 STATUS approved

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Last modified September 28 04:41 EDT 2023. Contains 365722 sequences. (Running on oeis4.)