A071403 - OEIS

%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,changed

%O 1,1

%A _Labos Elemer_, May 24 2002