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A066680
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Badly sieved numbers: as in the Sieve of Eratosthenes multiples of unmarked numbers p are marked, but only up to p^2.
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15
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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
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1,1
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COMMENTS
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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
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)
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LINKS
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Eric Weisstein's World of Mathematics, Sieve
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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PROG
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(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]
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CROSSREFS
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Sequences generated by a closely related sieving process: A000040 (also a subsequence), A026424, A270877.
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KEYWORD
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nonn,nice
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AUTHOR
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STATUS
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approved
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