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A066680 Badly sieved numbers: as in the Sieve of Eratosthenes multiples of unmarked numbers p are marked, but only up to p^2. 12
2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 23, 27, 29, 30, 31, 37, 41, 43, 45, 47, 50, 53, 59, 61, 63, 67, 70, 71, 73, 75, 79, 80, 83, 89, 97, 98, 101, 103, 105, 107, 109, 112, 113, 125, 127, 128, 131, 137, 139, 147, 149, 151, 154, 157, 163 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

A099104(a(n)) = 1.

a(A207432(n)) = A000040(n). [Reinhard Zumkeller, Feb 17 2012]

Obviously all primes and cubes of primes are in the sequence, while squares of primes are not. In fact, A000225 tells us which exponents prime powers in the sequence will exhibit.

But where it gets really interesting is in what happens to the Achilles numbers: the smallest badly sieved numbers that are also Achilles numbers are 864 and 972. - Alonso del Arte, Feb 21 2012

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

Eric Weisstein's World of Mathematics, Sieve

Wikipedia, Sieve theory

Index entries for sequences generated by sieves

EXAMPLE

For 2, the first unmarked number, there is only one multiple <= 4=2^2:

giving 2 3 [4] 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...

for 3, the next unmarked number, we mark 6=2*3 and 9=3*3

giving 2 3 [4] 5 [6] 7 8 [9] 10 11 12 13 14 15 16 17 18 19 20 ...

for 5, the next unmarked number, we mark 10=2*5, 15=3*5, 20=4*5 and 25=5*5

giving 2 3 [4] 5 [6] 7 8 [9] [10] 11 12 13 14 [15] 16 17 18 19 [20] ... and so on.

MATHEMATICA

A099104[1] = 0; A099104[n_] := A099104[n] = Product[If[n > d^2, 1, 1 - A099104[d]], {d, Select[ Range[n-1], Mod[n, #] == 0 &]}]; Select[ Range[200], A099104[#] == 1 &] (* Jean-Fran├žois Alcover, Feb 15 2012 *)

max = 200; badPrimes = Range[2, max]; len = max; iter = 1; While[iter <= len, curr = badPrimes[[iter]]; badPrimes = Complement[badPrimes, Range[2, curr]curr]; len = Length[badPrimes]; iter++]; badPrimes (* Alonso del Arte, Feb 21 2012 *)

PROG

(Haskell)

a066680 n = a066680_list !! (n-1)

a066680_list = s [2..] where

   s (b:bs) = b : s [x | x <- bs, x > b ^ 2 || mod x b > 0]

-- Reinhard Zumkeller, Feb 17 2012

CROSSREFS

Cf. A066681, A066682, A066683.

Cf. A099042, A099043.

Cf. A056875.

Sequence in context: A229125 A228853 A141832 * A298865 A211777 A189299

Adjacent sequences:  A066677 A066678 A066679 * A066681 A066682 A066683

KEYWORD

nonn,nice

AUTHOR

Reinhard Zumkeller, Dec 31, 2001

STATUS

approved

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Last modified February 19 18:30 EST 2019. Contains 320327 sequences. (Running on oeis4.)