A067514 -id:A067514 - cp's OEIS frontend

A055086 n appears 1+[n/2] times.

0, 1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16

Offset: 0

Michael Somos, Jun 13 2000

The PARI functions t1, t2 can be used to read a triangular array T(n,k) (n >= 0, 0 <= k <= floor(n/2)) by rows from left to right: n -> T(t1(n), t2(n)).
a(n) gives the number of distinct positive values taken by [n/k]. E.g., a(5)=3: [5/{1,2,3,4,5}]={5,2,1,1,1}. - Marc LeBrun, May 17 2001
This sequence gives the elements in increasing order of the set {i+2j} where i>=0, j>=0. - Benoit Cloitre, Sep 22 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Randell Heyman, Cardinality of a floor function set, arXiv:1905.00533 [math.NT], 2019.
Michael Somos, Sequences used for indexing triangular or square arrays, 2003.

Cf. A000267, A055087, A063524, A075993.

Cf. A067514.

Flatten[Table[Table[n,{Floor[n/2]+1}],{n,0,20}]] (* Harvey P. Dale, Mar 07 2014 *)

PARI
```
{a(n) = floor(sqrt(4*n + 1)) - 1}
```

t1(n)=floor(sqrt(1+4*n)-1) /* A055086 */

t2(n)=(1+4*n-sqr(floor(sqrt(1+4*n))))\4 /* A055087 */

a(n)=if(n<1,0,a(n-1-a(n-1)\2)+1) \\ Benoit Cloitre, May 09 2017

from math import isqrt
def A055086(n): return isqrt((n<<2)|1)-1 # Chai Wah Wu, Nov 23 2024

a(n) = [sqrt(4*n + 1)] - 1 = A000267(n) - 1.
a(n) = Sum_{k=1..n} A063524(A075993(n, k)), for n>0. - Reinhard Zumkeller, Apr 06 2006
a(n) = ceiling(2*sqrt(n+1)) - 2. - Mircea Merca, Feb 05 2012
a(0) = 0, then for n>=1 a(n) = 1 + a(n-1-floor(a(n-1)/2)). - Benoit Cloitre, May 08 2017
a(n) = floor(b) + floor(n/(floor(b)+1)) where b = (sqrt(4*n+1)-1)/2. - Randell G Heyman, May 08 2019
Sum_{k>=1} (-1)^(k+1)/a(k) = Pi/8 + 3*log(2)/4. - Amiram Eldar, Jan 26 2024

A068050 Number of values of k, 1<=k<=n, for which floor(n/k) is prime.

Original entry on oeis.org

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

Amarnath Murthy, Feb 12 2002

			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.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Randell Heyman, Primes in floor function sets, arXiv:2111.00408 [math.NT], 2021.

Cf. A067514.

Cf. A010051, A179119, A205745.

a068050 n = length [k | k <- [1..n], a010051 (n `div` k) == 1]
-- Reinhard Zumkeller, Jan 31 2012

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 *)

a(n) = sum(k=1, n, isprime(n\k)); \\ Michel Marcus, Jun 03 2024

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

Edited by Dean Hickerson, Feb 12 2002

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A055086 n appears 1+[n/2] times.

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A068050 Number of values of k, 1<=k<=n, for which floor(n/k) is prime.

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