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 A114205 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. 7
 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; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS 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. LINKS Table of n, a(n) for n=2..90. EXAMPLE 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 MAPLE 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, Oct 19 2006 MATHEMATICA fa[n_] := Block[{p}, p = First[RealDigits[1/n]]; If[ ! IntegerQ[Last[p]], p = Most[p]]; FromDigits[p]]; Table[fa[n], {n, 100}] (* Ray Chandler, Oct 18 2006 *) 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]]] CROSSREFS Cf. A114206, A036275, A051626, A060284, A007732. Sequence in context: A279603 A329252 A022665 * A316464 A167315 A167362 Adjacent sequences: A114202 A114203 A114204 * A114206 A114207 A114208 KEYWORD nonn,base AUTHOR N. J. A. Sloane, Oct 17 2006 EXTENSIONS More terms from Ray Chandler and Hans Havermann, Oct 18 2006 I would also like to get programs that produce this and A114206, A036275, A051626 in Maple. STATUS approved

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Last modified September 16 07:48 EDT 2024. Contains 375959 sequences. (Running on oeis4.)