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A085970
Number of integers ranging from 2 to n that are not prime-powers.
17
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
1,10
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
LINKS
FORMULA
a(n) = Max{A024619(k)<=n} k;
a(n) = n - A065515(n) = A085972(n) - A000720(n).
EXAMPLE
The a(30) = 13 numbers: 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30. - Gus Wiseman, Jul 28 2022
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 *)
PROG
(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
CROSSREFS
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.
Sequence in context: A285759 A365740 A082997 * A355538 A238884 A066683
KEYWORD
nonn
AUTHOR
Reinhard Zumkeller, Jul 06 2003
EXTENSIONS
Name modified by Gus Wiseman, Jul 28 2022. Normally 1 is not considered a prime-power, cf. A000961, A246655.
STATUS
approved