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A055399 Stages of sieve of Eratosthenes needed to identify n as prime or composite. 9
1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 3, 1, 2, 1, 3, 1, 3, 1, 2, 1, 3, 1, 3, 1, 2, 1, 4, 1, 4, 1, 2, 1, 3, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 4, 1, 4, 1, 2, 1, 5, 1, 3, 1, 2, 1, 5, 1, 5, 1, 2, 1, 3, 1, 5, 1, 2, 1, 5, 1, 5, 1, 2, 1, 4, 1, 5, 1, 2, 1, 5, 1, 3, 1, 2, 1, 5 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,3

COMMENTS

Primes are known as primes actually one step before a(n): at step k of the sieve, multiples of prime(k) are removed, the smallest integer removed being prime(k)^2; every remaining integer less than prime(k+1)^2 will then never be removed, and it is newly known at step k for those between prime(k)^2 and prime(k+1)^2. For example, at step 3, multiples of prime(3) = 5 are removed and remaining integers after this step are prime up to prime(4)^2 = 49; then, 29, 31, 37, 41, 43, 47 are known as prime at step 3. - Jean-Christophe Hervé, Nov 01 2013

LINKS

T. D. Noe, Table of n, a(n) for n = 3..10000

H. B. Meyer, Eratosthenes' sieve

J. Britton, Sieve of Eratosthenes Applet

C. K. Caldwell, The Prime Glossary, sieve of Eratosthenes

FORMULA

If n is composite, a(n) = A055396(n); if n is prime, a(n) = pi(firstprimeabove(sqrt(n)). [Corrected by Charles R Greathouse IV, Sep 03 2013]

a(n) = A010051(n)*(A056811(n)+1)+(1-A010051(n))*A055396(n). - Jean-Christophe Hervé, Nov 01 2013

EXAMPLE

a(7)=2 because 7 is not removed by the first two stages of the sieve, but is less than the square of the second prime (though not the square of the first); a(35)=3 because 35 is removed in the third stage as a multiple of 5.

MATHEMATICA

a[n_ /; !PrimeQ[n]] := PrimePi[ FactorInteger[n][[1, 1]]]; a[n_ /; PrimeQ[n]] := PrimePi[ NextPrime[ Sqrt[n]]]; Table[a[n], {n, 3, 107}](* Jean-François Alcover, Jun 11 2012, after formula *)

CROSSREFS

Cf. A000040, A002808, A004280, A038179, A055396, A054403, A055398, A083269, A010051, A056811.

Sequence in context: A302776 A060775 A175494 * A029426 A085342 A025825

Adjacent sequences:  A055396 A055397 A055398 * A055400 A055401 A055402

KEYWORD

nice,nonn,easy

AUTHOR

Henry Bottomley, May 15 2000

STATUS

approved

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Last modified August 24 03:51 EDT 2019. Contains 326260 sequences. (Running on oeis4.)