A085970 Number of integers ranging from 2 to n that are not prime-powers.

0, 0, 0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 5, 5, 6, 6, 7, 8, 9, 9, 10, 10, 11, 11, 12, 12, 13, 13, 13, 14, 15, 16, 17, 17, 18, 19, 20, 20, 21, 21, 22, 23, 24, 24, 25, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 40, 41, 41, 42, 42, 43

Offset: 1

Author

Reinhard Zumkeller, Jul 06 2003

Keywords

• nonn

Comments

For n > 2, a(n) gives the number of duplicate eliminations performed by the Sieve of Eratosthenes when sieving the interval [2, n]. - Felix Fröhlich, Dec 10 2016

Number of terms of A024619 <= n. - Felix Fröhlich, Dec 10 2016

First differs from A082997 at n = 30. - Gus Wiseman, Jul 28 2022

Examples

			The a(30) = 13 numbers: 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30. - _Gus Wiseman_, Jul 28 2022
		

Links

• Felix Fröhlich, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000720, A024619, A085972.

The complement is counted by A065515, without 1's A025528.

For primes instead of prime-powers we have A065855, with 1's A062298.

Partial sums of A143731.

The version not treating 1 as a prime-power is A356068.

A000688 counts factorizations into prime-powers.

A001222 counts prime-power divisors.

A246655 lists the prime-powers (A000961 includes 1), towers A164336.

Cf. A023893, A023894, A036234, A069513, A207375, A354911, A355743, A356064, A356065.

Programs

Mathematica

With[{nn = 75}, Table[n - Count[#, k_ /; k < n] - 1, {n, nn}] &@ Join[{1}, Select[Range@ nn, PrimePowerQ]]] (* Michael De Vlieger, Dec 11 2016 *)

PARI

a(n) = my(i=0); forcomposite(c=4, n, if(!isprimepower(c), i++)); i \\ Felix Fröhlich, Dec 10 2016

Python

from sympy import primepi, integer_nthroot
def A085970(n): return n-1-sum(primepi(integer_nthroot(n,k)[0]) for k in range(1,n.bit_length())) # Chai Wah Wu, Aug 20 2024

Formula

a(n) = Max{A024619(k)<=n} k;

a(n) = n - A065515(n) = A085972(n) - A000720(n).

Extensions

Name modified by Gus Wiseman, Jul 28 2022. Normally 1 is not considered a prime-power, cf. A000961, A246655.