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

1,2

Comments

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

a(n) = A175046(A000040(n)). - Reinhard Zumkeller, Apr 05 2010

Links

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

Formula

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

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

Example

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.

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);

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A288625 A189168 A182799 * A242324 A099519 A014121

Adjacent sequences: A112926 A112927 A112928 * A112930 A112931 A112932

Keyword

nonn

Author

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

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

Mathematica syntax corrected by Frank M Jackson, Dec 28 2018

Status

approved