A067514 Number of distinct primes of the form floor(n/k) for 1 <= k <= n.
0, 1, 1, 1, 2, 2, 3, 1, 2, 3, 4, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 5, 6, 2, 3, 4, 4, 4, 5, 4, 5, 3, 4, 5, 6, 4, 5, 5, 6, 4, 5, 4, 5, 5, 5, 6, 7, 3, 4, 4, 5, 6, 7, 5, 6, 5, 6, 7, 8, 4, 5, 5, 5, 4, 5, 6, 7, 7, 8, 7, 8, 4, 5, 5, 5, 5, 6, 7, 8, 6, 6, 7, 8, 4, 5, 6, 7, 7, 8, 5, 6, 7, 8, 9, 10, 6, 7, 5, 6, 5, 6, 6
Offset: 1
Examples
a(10)=3 as floor(10/k) for k = 1 to 10 is 10,5,3,2,2,1,1,1,1,1, respectively; the 3 primes are 5,3,2.
Links
- G. C. Greubel, Table of n, a(n) for n = 1..1000
- Randell Heyman, Cardinality of a floor function, arXiv:1905.00533 [math.NT], 2019.
- Randell Heyman, Primes in floor function sets, arXiv:2111.00408 [math.NT], 2021
- Randell Heyman, Primes in floor function sets, INTEGERS, vol. 22, 2022, A59.
Crossrefs
Cf. A068050.
Cf. A055086 (number of distinct integers with same form). - Michel Marcus, May 04 2019
Programs
-
Mathematica
a[n_] := Length[Union[Select[Table[Floor[n/i], {i, 1, n}], PrimeQ]]] Table[PrimeNu[Product[Floor[n/k], {k, 1, n}]], {n, 1, 100}] (* G. C. Greubel, May 08 2017 *) -
PARI
a(n) = #select(x->isprime(x), Set(vector(n, k, n\k))); \\ Michel Marcus, May 04 2019
-
PARI
a(n)=my(s=sqrtint(n+1)); sum(k=1,s,isprime(n\k))+primepi(n\s-1) \\ Charles R Greathouse IV, Nov 05 2021
Formula
a(n) = 4*n^(1/2)/log(n) + O(n^(1/2)/(log(n))^2). - Randell Heyman, Oct 06 2022
Extensions
Edited by Dean Hickerson, Feb 12 2002