

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
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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.  JeanChristophe 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)+(1A010051(n))*A055396(n).  JeanChristophe 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}](* JeanFrançois 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



