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Search: id:A114205
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| A114205 |
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Write decimal expansion of 1/n as 0.PPP...PQQQ..., where QQQ... is the cyclic part. If the expansion does not terminate, any leading 0's in QQQ... are regarded as being at the end of the PPP...P part. Sequence gives PPP...P, right justified, with leading zeros omitted. |
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3
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| 5, 0, 25, 2, 1, 0, 125, 0, 1, 0, 8, 0, 0, 0, 625, 0, 0, 0, 5, 0, 0, 0, 41, 4, 0, 0, 3, 0, 0, 0, 3125, 0, 0, 0, 2, 0, 0, 0, 25, 0, 0, 0, 2, 0, 0, 0, 208, 0, 2, 0, 1, 0, 0, 0, 17, 0, 0, 0, 1, 0, 0, 0, 15625, 0, 0, 0, 1, 0, 0, 0, 13, 0, 0, 1, 1, 0, 0, 0, 125, 0, 0, 0, 1, 0, 0, 0, 11, 0, 0
(list; graph; listen)
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OFFSET
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2,1
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COMMENT
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b(n) = A114206(n) gives the length of P (including leading zeros), c(n) = A036275(n) gives the smallest cycle in QQQ... (including terminating zeros), and d(n) = A051626(n) gives the length of that cycle.
Thus 1/n = 10^(-b(n)) * ( a(n) + c(n)/(10^d(n) - 1) ). When c(n)=d(n)=0, the fraction c(n)/(10^d(n) - 1), which is 0/0, evaluates (by definition) to 0.
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EXAMPLE
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n .. expansion of 1/n .... a b c d
2 .50000000000000000000... 5 1 0 0
3 .33333333333333333333... 0 0 3 1
4 .25000000000000000000... 25 2 0 0
5 .20000000000000000000... 2 1 0 0
6 .16666666666666666667... 1 1 6 1
7 .14285714285714285714... 0 0 142857 6
8 .12500000000000000000... 125 3 0 0
9 .11111111111111111111... 0 0 1 1
10 .1000000000000000000... 1 1 0 0
11 .0909090909090909090... 0 1 90 2
12 .0833333333333333333... 8 2 3 1
13 .0769230769230769230... 0 1 769230 6
14 .0714285714285714285... 0 1 714285 6
15 .0666666666666666666... 0 1 6 1
16 .0625000000000000000... 625 4 0 0
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MAPLE
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A114205 := proc(n) local sh, lpow, mpow, a, b ; lpow:=1 ; while true do for mpow from lpow-1 to 0 by -1 do if (10^lpow-10^mpow) mod n =0 then a := (10^lpow-10^mpow)/n ; sh := 10^(lpow-mpow)-1 ; b := a mod sh ; a := floor(a/sh) ; while b>0 and b*10 < sh+1 do a := 10*a ; b := 10*b ; end ; RETURN(a) ; fi ; od ; lpow := lpow+1 ; od ; end: for n from 2 to 600 do printf("%d %d ", n, A114205(n)) ; od ; - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 19 2006
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MATHEMATICA
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fa[n_] := Block[{p}, p = First[RealDigits[1/n]]; If[ ! IntegerQ[Last[p]], p = Most[p]]; FromDigits[p]]; Table[fa[n], {n, 100}] (*Chandler*)
Mathematica code from Hans Havermann, Oct 19 2006:
r[x_] := RealDigits[1/x]
w[x_] := First[r[x]]
f[x_] := First[w[x]]
l[x_] := Last[w[x]]
z[x_] := Last[r[x]]
a[x_] := Which[IntegerQ[l[x]], FromDigits[w[x]], IntegerQ[f[x]] ==False, 0, True, FromDigits[Drop[w[x], -1]]]
b[x_] := Which[IntegerQ[l[x]], Length[w[x]]-1*z[x], IntegerQ[f[x]] ==False, -1*z[x], True, Length[Drop[w[x], -1]]-1*z[x]]
c[x_] := Which[IntegerQ[l[x]], 0, IntegerQ[f[x]]==False, FromDigits[f[x]], True, FromDigits[l[x]]]
d[x_] := Which[IntegerQ[l[x]], 0, IntegerQ[f[x]]==False, Length[f[x]], True, Length[l[x]]]
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CROSSREFS
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Cf. A114206, A036275, A051626, A060284, A007732.
Sequence in context: A094031 A056461 A022665 this_sequence A070611 A047769 A103709
Adjacent sequences: A114202 A114203 A114204 this_sequence A114206 A114207 A114208
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KEYWORD
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nonn
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AUTHOR
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njas, Oct 17 2006
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EXTENSIONS
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More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net) and Hans Havermann, Oct 18 2006
I would also like to get programs that produce this and A114206, A036275, A051626 in Maple.
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