OFFSET
1,2
COMMENTS
a(n) = order of n-th term of A112925 among squarefree integers.
LINKS
Diana Mecum and Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 200 terms from Mecum)
FORMULA
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);
# 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 *)
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
(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
CROSSREFS
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
STATUS
approved