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.

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, 0, 10

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..92.

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
(Start)
92 .0108695652173913043... 10 3(?) 869...260 22
102 .009803921568627450... 0 2 980...450 16
416 .002403846153846153... 240 5 384615 6
4544 .00022007042253521... 2200 7(?) 704...450 35
(End) - Ruud H.G. van Tol, Nov 20 2024

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 *)
(* alternate program *)
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]]] (* Hans Havermann, Oct 19 2006 *)

Prog

(PARI) a(n)= my(s=max(valuation(n, 2), valuation(n, 5))); s||return(0); my([p, r]= divrem(10^s, n)); if(r&&(r=n\r)>9, s+=logint(r, 10)); 10^s\n; \\ Ruud H.G. van Tol, Nov 19 2024

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