A068050 -id:A068050 - cp's OEIS frontend

A068007 Least number k such that the number of primes of the form [k/j] for j=1..k (A068050) is n, or zero if impossible.

1, 2, 5, 7, 10, 11, 15, 21, 0, 22, 23, 31, 34, 38, 35, 45, 50, 46, 47, 53, 62, 58, 59, 67, 69, 84, 70, 71, 79, 83, 87, 92, 93, 101, 94, 105, 95, 106, 107, 116, 117, 122, 118, 125, 119, 134, 135, 139, 146, 142, 149, 143, 156, 155, 158, 159, 171, 166, 167, 176, 175, 185

Offset: 0

Robert G. Wilson v, Feb 12 2002

nonn

a(n) = 0 for n = 8, 94, 103, 122, 180, 283, 311, 353, 355, 398, ...

Cf. A068050.

f[n_] := Count[ PrimeQ[ Floor[ n/Table[i, {i, Floor[n/2]} ]]], True]; a = Table[0, {100} ]; Do[b = f[n]; If[b < 100 && a[[b + 1]] == 0, a[[b + 1]] = n], {n, 1, 300}]; a

Name corrected by Sean A. Irvine, Jan 18 2024

A205745 a(n) = card { d | d*p = n, d odd, p prime }.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 1, 1, 2, 0, 2, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 2, 1, 1, 0, 1, 1, 2, 0, 2, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2

Offset: 1

Peter Luschny, Jan 30 2012

nonn

Equivalently, a(n) is the number of prime divisors p|n such that n/p is odd. - Gus Wiseman, Jun 06 2018

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A000005, A000607, A001221, A008683, A010051, A068050, A077761, A083399, A088705, A106404, A305614.

a205745 n = sum $ map ((`mod` 2) . (n `div`))
   [p | p <- takeWhile (<= n) a000040_list, n `mod` p == 0]
-- Reinhard Zumkeller, Jan 31 2012

a[n_] := Sum[ Boole[ OddQ[d] && PrimeQ[n/d] ], {d, Divisors[n]} ]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 27 2013 *)

a(n)=if(n%2,omega(n),n%4/2) \\ Charles R Greathouse IV, Jan 30 2012

def A205745(n):
    return sum((n//d) % 2 for d in divisors(n) if is_prime(d))
[A205745(n) for n in (1..105)]

O.g.f.: Sum_{p prime} x^p/(1 - x^(2p)). - Gus Wiseman, Jun 06 2018

Sum_{k=1..n} a(k) = (n/2) * (log(log(n)) + B) + O(n/log(n)), where B is Mertens's constant (A077761). - Amiram Eldar, Sep 21 2024

A067514 Number of distinct primes of the form floor(n/k) for 1 <= k <= n.

Original entry on oeis.org

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

Amarnath Murthy, Feb 12 2002

			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.

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.

Cf. A068050.

Cf. A055086 (number of distinct integers with same form). - Michel Marcus, May 04 2019

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 *)

a(n) = #select(x->isprime(x), Set(vector(n, k, n\k))); \\ Michel Marcus, May 04 2019

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

a(n) = A001221(A010786(n)). - Enrique Pérez Herrero, Feb 26 2012

a(n) = 4*n^(1/2)/log(n) + O(n^(1/2)/(log(n))^2). - Randell Heyman, Oct 06 2022

Edited by Dean Hickerson, Feb 12 2002

cp's OEIS Frontend

A068007 Least number k such that the number of primes of the form [k/j] for j=1..k (A068050) is n, or zero if impossible.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Extensions

A205745 a(n) = card { d | d*p = n, d odd, p prime }.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Sage

Formula

A067514 Number of distinct primes of the form floor(n/k) for 1 <= k <= n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions