A066680 Badly sieved numbers: as in the Sieve of Eratosthenes multiples of unmarked numbers p are marked, but only up to p^2.

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

Offset: 1

Author

Reinhard Zumkeller, Dec 31 2001

Keywords

• nonn
• nice

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 3-almost prime is present if and only if its largest prime factor is less than its square root. The first term that is a 4-almost prime is 220.

The effect of this sieve can be compared against the A270877 trapezoidal sieve. Each unmarked number k marks k-1 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)

Examples

			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.
		

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

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.

Programs

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

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-François 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 *)