|
%I #41 Nov 15 2022 11:11:44
%S 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,
%T 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,
%U 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
%N Number of distinct primes of the form floor(n/k) for 1 <= k <= n.
%H G. C. Greubel, <a href="/A067514/b067514.txt">Table of n, a(n) for n = 1..1000</a>
%H Randell Heyman, <a href="https://arxiv.org/abs/1905.00533">Cardinality of a floor function</a>, arXiv:1905.00533 [math.NT], 2019.
%H Randell Heyman, <a href="https://arxiv.org/abs/2111.00408">Primes in floor function sets</a>, arXiv:2111.00408 [math.NT], 2021
%H Randell Heyman, <a href="/A067514/a067514.pdf">Primes in floor function sets</a>, INTEGERS, vol. 22, 2022, A59.
%F a(n) = A001221(A010786(n)). - _Enrique Pérez Herrero_, Feb 26 2012
%F a(n) = 4*n^(1/2)/log(n) + O(n^(1/2)/(log(n))^2). - _Randell Heyman_, Oct 06 2022
%e 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.
%t a[n_] := Length[Union[Select[Table[Floor[n/i], {i, 1, n}], PrimeQ]]]
%t Table[PrimeNu[Product[Floor[n/k], {k, 1, n}]], {n, 1, 100}] (* _G. C. Greubel_, May 08 2017 *)
%o (PARI) a(n) = #select(x->isprime(x), Set(vector(n, k, n\k))); \\ _Michel Marcus_, May 04 2019
%o (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
%Y Cf. A068050.
%Y Cf. A055086 (number of distinct integers with same form). - _Michel Marcus_, May 04 2019
%K easy,nonn
%O 1,5
%A _Amarnath Murthy_, Feb 12 2002
%E Edited by _Dean Hickerson_, Feb 12 2002
|
