A112929 Number of squarefree integers less than the n-th prime.

1, 2, 3, 5, 7, 8, 11, 12, 15, 17, 19, 23, 26, 28, 30, 32, 36, 37, 41, 44, 45, 49, 51, 55, 60, 61, 63, 66, 67, 70, 77, 80, 83, 85, 91, 92, 95, 99, 102, 104, 108, 109, 116, 117, 120, 121, 129, 138, 140, 141, 144, 148, 149, 153, 157, 161, 165, 166, 169, 171, 173, 179, 187

Offset: 1

Author

Leroy Quet, Oct 06 2005 and Emeric Deutsch, Oct 14 2005

Keywords

• nonn

Comments

a(n) = order of n-th term of A112925 among squarefree integers.

Examples

			a(5)=7 because the 5th prime is 11 and the squarefree numbers not exceeding 11 are: 2,3,5,6,7,10,11.
The 5th term of A112925 is 10 and 10 is the 7th squarefree integer (with 1 counted as the first squarefree integer). So a(5) = 7.
		

Links

• Diana Mecum and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 200 terms from Mecum)

Crossrefs

Cf. A013928, A112925, A112926, A061400, A112928, A112930.

Programs

Maple

with(numtheory): a:=proc(n) local p,B,j: p:=ithprime(n): B:={}: for j from 2 to p do if abs(mobius(j))>0 then B:=B union {j} else B:=B fi od: nops(B) end: seq(a(m),m=1..75);
# Or:
a := n -> nops(select(NumberTheory:-IsSquareFree, [seq(1..ithprime(n)-1)])):
seq(a(n), n=1..63);  # Peter Luschny, Dec 12 2024

Mathematica

f[n_] := Prime[n] - Sum[ If[ MoebiusMu[k]==0, 1, 0], {k, Prime[n]}] - 1; Table[ f[n], {n, 63}] (* Robert G. Wilson v, Oct 15 2005; syntax corrected by Frank M Jackson, Dec 28 2018 *)

PARI

a(n)={
my(lim=prime(n)-1,b=sqrtint(lim\2));
sum(k=1,b,moebius(k)*(lim\k^2))+
sum(k=b+1,sqrt(lim),moebius(k))
}; \\ Charles R Greathouse IV, Apr 26 2012

PARI

a(n,p=prime(n))=p--; my(s,b=sqrtint(p\2)); forsquarefree(k=1, b, s += p\k[1]^2*moebius(k)); forsquarefree(k=b+1, sqrtint(p), s += moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018

Python

from math import isqrt
from sympy import prime, mobius
def A112929(n): return (p:=prime(n))-1+sum(mobius(k)*(p//k**2) for k in range(2,isqrt(p)+1)) # Chai Wah Wu, Dec 12 2024

Formula

A005117(a(n)) = A112925(n). - R. J. Mathar, Apr 19 2008

a(n) = A013928(A000040(n)). - Reinhard Zumkeller, Apr 05 2010

a(n) ~ 6/Pi^2 * n log n. - Charles R Greathouse IV, Apr 26 2012

Extensions

More terms from Diana L. Mecum, May 29 2007

Edited by N. J. A. Sloane, Apr 26 2008 at the suggestion of R. J. Mathar