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%I #29 Dec 28 2018 14:11:46
%S 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,
%T 61,63,66,67,70,77,80,83,85,91,92,95,99,102,104,108,109,116,117,120,
%U 121,129,138,140,141,144,148,149,153,157,161,165,166,169,171,173,179,187
%N Number of squarefree integers less than the n-th prime.
%C a(n) = order of n-th term of A112925 among squarefree integers.
%C a(n) = A175046(A000040(n)). - _Reinhard Zumkeller_, Apr 05 2010
%H Diana Mecum and Charles R Greathouse IV, <a href="/A112929/b112929.txt">Table of n, a(n) for n = 1..10000</a> (first 200 terms from Mecum)
%F A005117(a(n)) = A112925(n). - _R. J. Mathar_, Apr 19 2008
%F a(n) ~ 6/Pi^2 * n log n. - _Charles R Greathouse IV_, Apr 26 2012
%e a(5)=7 because the 5th prime is 11 and the squarefree numbers not exceeding 11 are: 2,3,5,6,7,10,11.
%e 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.
%p 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);
%t 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 *)
%o (PARI) a(n)={
%o my(lim=prime(n)-1,b=sqrtint(lim\2));
%o sum(k=1,b,moebius(k)*(lim\k^2))+
%o sum(k=b+1,sqrt(lim),moebius(k))
%o }; \\ _Charles R Greathouse IV_, Apr 26 2012
%o (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
%Y Cf. A112925, A112926, A061400, A112928, A112930.
%K nonn
%O 1,2
%A _Leroy Quet_, Oct 06 2005 and _Emeric Deutsch_, Oct 14 2005
%E More terms from _Diana L. Mecum_, May 29 2007
%E Edited by _N. J. A. Sloane_, Apr 26 2008 at the suggestion of _R. J. Mathar_
%E Mathematica syntax corrected by _Frank M Jackson_, Dec 28 2018
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