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%I #25 Feb 26 2024 02:20:33
%S 1,7,61,607,6077,60787,607951,6079284,60792732,607927092,6079270913,
%T 60792710227,607927101577,6079271018873,60792710185938,
%U 607927101853650,6079271018542500,60792710185398417,607927101854027370,6079271018540264581,60792710185402679735,607927101854026683706
%N Number of squarefree integers in the closed interval [10^n, -1 + 2*10^n], i.e., among 10^n consecutive integers beginning with 10^n.
%C What a(n)/n is converging to?
%C Limit_{n->oo} a(n)/10^n = 1/zeta(2). [_Max Alekseyev_, Oct 18 2008]
%H Lucas A. Brown, <a href="https://github.com/lucasaugustus/oeis/blob/main/A077642.py">Python program</a>.
%F a(n) = Sum_{j=0..-1+10^n} abs(mu(10^n + j)).
%e n=10: among numbers {10,...,19} seven are squarefree [10,11,13,14,15,17,19], so a(1)=7.
%p with(numtheory): for n from 0 to 5 do ct:=0: for k from 10^n to 2*10^n-1 do if abs(mobius(k))>0 then ct:=ct+1 else ct:=ct: fi: od: a[n]:=ct: od: seq(a[n],n=0..5); # _Emeric Deutsch_, Mar 28 2005
%t Table[Apply[Plus, Table[Abs[MoebiusMu[10^w+j]], {j, 0, -1+10^(w-1)}]], {w, 0, 6}]
%o (PARI) { a(n) = sum(m=1,sqrtint(2*10^n-1), moebius(m) * ((2*10^n-1)\m^2 - (10^n-1)\m^2) ) } \\ _Max Alekseyev_, Oct 18 2008
%Y Cf. A077641, A077643.
%K nonn
%O 0,2
%A _Labos Elemer_, Nov 14 2002
%E 6079284 from _Emeric Deutsch_, Mar 28 2005
%E a(8)-a(15) from _Max Alekseyev_, Oct 18 2008
%E a(16)-a(21) from _Lucas A. Brown_, Feb 25 2024
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