

A112929


Number of squarefree integers less than the nth prime.


7



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
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OFFSET

1,2


COMMENTS

a(n) = order of nth 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



