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A077642
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.
2
1, 7, 61, 607, 6077, 60787, 607951, 6079284, 60792732, 607927092, 6079270913, 60792710227, 607927101577, 6079271018873, 60792710185938, 607927101853650, 6079271018542500, 60792710185398417, 607927101854027370, 6079271018540264581, 60792710185402679735, 607927101854026683706
OFFSET
0,2
COMMENTS
What a(n)/n is converging to?
Limit_{n->oo} a(n)/10^n = 1/zeta(2). [Max Alekseyev, Oct 18 2008]
FORMULA
a(n) = Sum_{j=0..-1+10^n} abs(mu(10^n + j)).
EXAMPLE
n=10: among numbers {10,...,19} seven are squarefree [10,11,13,14,15,17,19], so a(1)=7.
MAPLE
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
MATHEMATICA
Table[Apply[Plus, Table[Abs[MoebiusMu[10^w+j]], {j, 0, -1+10^(w-1)}]], {w, 0, 6}]
PROG
(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
CROSSREFS
Sequence in context: A113718 A177132 A364430 * A071172 A259335 A127688
KEYWORD
nonn
AUTHOR
Labos Elemer, Nov 14 2002
EXTENSIONS
6079284 from Emeric Deutsch, Mar 28 2005
a(8)-a(15) from Max Alekseyev, Oct 18 2008
a(16)-a(21) from Lucas A. Brown, Feb 25 2024
STATUS
approved