A068050 Number of values of k, 1<=k<=n, for which floor(n/k) is prime.
0, 1, 1, 1, 2, 2, 3, 2, 2, 4, 5, 3, 4, 5, 6, 5, 6, 5, 6, 6, 7, 9, 10, 6, 7, 9, 9, 9, 10, 10, 11, 9, 10, 12, 14, 11, 12, 13, 14, 13, 14, 13, 14, 14, 15, 17, 18, 13, 14, 16, 17, 18, 19, 17, 19, 18, 19, 21, 22, 18, 19, 20, 21, 19, 21, 22, 23, 23, 24, 26, 27, 21, 22, 23, 24, 24, 26, 27
Offset: 1
Examples
a(10) = 4 as floor(10/k) for k = 1 to 10 is 10,5,3,2,2,1,1,1,1,1, respectively; this is prime for k = 2,3,4,5.
Links
- Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
- Randell Heyman, Primes in floor function sets, arXiv:2111.00408 [math.NT], 2021.
Programs
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Haskell
a068050 n = length [k | k <- [1..n], a010051 (n `div` k) == 1] -- Reinhard Zumkeller, Jan 31 2012
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Mathematica
a[n_] := Length[Select[Table[Floor[n/i], {i, 1, n}], PrimeQ]] Table[Count[Table[Floor[n/k],{k,n}],?PrimeQ],{n,80}] (* _Harvey P. Dale, Nov 19 2022 *) -
PARI
a(n) = sum(k=1, n, isprime(n\k)); \\ Michel Marcus, Jun 03 2024
Formula
If p is a prime other than 3, a(p) = a(p-1) + 1. - Franklin T. Adams-Watters, Apr 27 2020
a(n) = A179119*n + O(n^(1/2)). - Randell Heyman, Oct 06 2022
a(n) = Sum_{p prime and p<=n} (floor(n/p) - floor(n/(p+1))). - Ridouane Oudra, Jun 03 2024
Extensions
Edited by Dean Hickerson, Feb 12 2002