%I #31 Dec 09 2024 19:45:16

%S 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,

%T 62,64,67,68,71,78,81,84,86,92,93,96,100,103,105,109,110,117,118,121,

%U 122,130,139,141,142,145,149,150,154,158,162,166,167,170,172,174,180

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

%C Also the number of squarefree numbers <= prime(n). - _Gus Wiseman_, Dec 08 2024

%H Charles R Greathouse IV, <a href="/A071403/b071403.txt">Table of n, a(n) for n = 1..10000</a>

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

%F a(n) ~ (6/Pi^2) * n log n. - _Charles R Greathouse IV_, Nov 27 2017

%F a(n) = A013928(A000040(n)). - _Ridouane Oudra_, Oct 15 2019

%F From _Gus Wiseman_, Dec 08 2024: (Start)

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

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

%F (End)

%e a(25)=61 because A005117(61) = prime(25) = 97.

%e From _Gus Wiseman_, Dec 08 2024: (Start)

%e The squarefree numbers up to prime(n) begin:

%e n = 1 2 3 4 5 6 7 8 9 10

%e ----------------------------------

%e 2 3 5 7 11 13 17 19 23 29

%e 1 2 3 6 10 11 15 17 22 26

%e 1 2 5 7 10 14 15 21 23

%e 1 3 6 7 13 14 19 22

%e 2 5 6 11 13 17 21

%e 1 3 5 10 11 15 19

%e 2 3 7 10 14 17

%e 1 2 6 7 13 15

%e 1 5 6 11 14

%e 3 5 10 13

%e 2 3 7 11

%e 1 2 6 10

%e 1 5 7

%e 3 6

%e 2 5

%e 1 3

%e 2

%e 1

%e The column-lengths are a(n).

%e (End)

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

%o (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

%o (PARI) a(n,p=prime(n))=sum(k=1, sqrtint(p), p\k^2*moebius(k)) \\ _Charles R Greathouse IV_, Sep 13 2013

%o (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

%o (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

%o (Python)

%o from math import isqrt

%o from sympy import prime, mobius

%o 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

%Y Cf. A000290, A013928.

%Y The strict version is A112929.

%Y A000040 lists the primes, differences A001223, seconds A036263.

%Y A005117 lists the squarefree numbers, differences A076259.

%Y A013929 lists the nonsquarefree numbers, differences A078147.

%Y A070321 gives the greatest squarefree number up to n.

%Y Other families: A014689, A027883, A378615, A065890.

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

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

%Y Cf. A046933, A049093, A053797, A072284, A077641, A224363, A337030, A345531.

%K nonn

%O 1,1

%A _Labos Elemer_, May 24 2002