A071403 Which squarefree number is prime? a(n)-th squarefree number equals n-th prime.

2, 3, 4, 6, 8, 9, 12, 13, 16, 18, 20, 24, 27, 29, 31, 33, 37, 38, 42, 45, 46, 50, 52, 56, 61, 62, 64, 67, 68, 71, 78, 81, 84, 86, 92, 93, 96, 100, 103, 105, 109, 110, 117, 118, 121, 122, 130, 139, 141, 142, 145, 149, 150, 154, 158, 162, 166, 167, 170, 172, 174, 180

Offset: 1

Author

Labos Elemer, May 24 2002

Keywords

• nonn

Comments

Also the number of squarefree numbers <= prime(n). - Gus Wiseman, Dec 08 2024

Examples

			a(25)=61 because A005117(61) = prime(25) = 97.
From _Gus Wiseman_, Dec 08 2024: (Start)
The squarefree numbers up to prime(n) begin:
n = 1  2  3  4   5   6   7   8   9  10
    ----------------------------------
    2  3  5  7  11  13  17  19  23  29
    1  2  3  6  10  11  15  17  22  26
       1  2  5   7  10  14  15  21  23
          1  3   6   7  13  14  19  22
             2   5   6  11  13  17  21
             1   3   5  10  11  15  19
                 2   3   7  10  14  17
                 1   2   6   7  13  15
                     1   5   6  11  14
                         3   5  10  13
                         2   3   7  11
                         1   2   6  10
                             1   5   7
                                 3   6
                                 2   5
                                 1   3
                                     2
                                     1
The column-lengths are a(n).
(End)
		

Links

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

Crossrefs

Cf. A000290, A013928.

The strict version is A112929.

A000040 lists the primes, differences A001223, seconds A036263.

A005117 lists the squarefree numbers, differences A076259.

A013929 lists the nonsquarefree numbers, differences A078147.

A070321 gives the greatest squarefree number up to n.

Other families: A014689, A027883, A378615, A065890.

Squarefree numbers between primes: A061398, A068360, A373197, A373198, A377430, A112925, A112926.

Nonsquarefree numbers: A057627, A378086, A061399, A068361, A120327, A377783, A378032, A378033.

Cf. A046933, A049093, A053797, A072284, A077641, A224363, A337030, A345531, A008864.

Programs

Mathematica

Position[Select[Range[300], SquareFreeQ], ?PrimeQ][[All, 1]] (* _Michael De Vlieger, Aug 17 2023 *)

PARI

lista(nn)=sqfs = select(n->issquarefree(n), vector(nn, i, i)); for (i = 1, #sqfs, if (isprime(sqfs[i]), print1(i, ", "));); \\ Michel Marcus, Sep 11 2013

PARI

a(n,p=prime(n))=sum(k=1, sqrtint(p), p\k^2*moebius(k)) \\ Charles R Greathouse IV, Sep 13 2013

PARI

a(n,p=prime(n))=my(s); forfactored(k=1, sqrtint(p), s+=p\k[1]^2*moebius(k)); s \\ Charles R Greathouse IV, Nov 27 2017

PARI

first(n)=my(v=vector(n),pr,k); forsquarefree(m=1,n*logint(n,2)+3, k++; if(m[2][,2]==[1]~, v[pr++]=k; if(pr==n, return(v)))) \\ Charles R Greathouse IV, Jan 08 2018

Python

from math import isqrt
from sympy import prime, mobius
def A071403(n): return (p:=prime(n))+sum(mobius(k)*(p//k**2) for k in range(2,isqrt(p)+1)) # Chai Wah Wu, Jul 20 2024

Formula

A005117(a(n)) = A000040(n) = prime(n).

a(n) ~ (6/Pi^2) * n log n. - Charles R Greathouse IV, Nov 27 2017

a(n) = A013928(A008864(n)). - Ridouane Oudra, Oct 15 2019

From Gus Wiseman, Dec 08 2024: (Start)

a(n) = A112929(n) + 1.

a(n+1) - a(n) = A373198(n) = A061398(n) - 1.

(End)