

A066680


Badly sieved numbers: as in the Sieve of Eratosthenes multiples of unmarked numbers p are marked, but only up to p^2.


13



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
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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
From Peter Munn, Aug 09 2019: (Start)
The factorization pattern of a number's divisors (as defined in A191743) determines whether a number is a term.
There are no semiprimes in the sequence, and a 3almost prime is present if and only if its largest prime factor is less than its square root. The first term that is a 4almost prime is 220.
The effect of this sieve can be compared against the A270877 trapezoidal sieve. Each unmarked number k marks k1 numbers in both sieves; but the largest number marked by k in this sieve is k^2, about twice the largest number marked by k in A270877 (the triangular number T_k = k(k+1)/2). The relative densities early in the two sequences are illustrated by a(10) = 18 < A270877(10) = 19, a(100) = 312 > A270877(100) = 268, a(1000) = 4297 > A270877(1000) = 2894.
(End)


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[n1], Mod[n, #] == 0 &]}]; Select[ Range[200], A099104[#] == 1 &] (* JeanFrançois 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 !! (n1)
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

A066681, A066682, A066683, A099042, A099043, A207432 have analysis of this sequence.
Cf. A056875, A075362, A099104 (characteristic function), A191743.
Sequences generated by a closely related sieving process: A000040 (also a subsequence), A026424, A270877.
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



