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

Author

N. J. A. Sloane, Oct 17 2006

Keywords

• nonn
• base

Comments

b(n) = A386406(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.

Examples

			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
		

Crossrefs

Cf. A386406, A114206, A036275, A051626, A060284, A007732, A175555.

Programs

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 =  Join[Most[p],TakeWhile[Last[p],#==0&]]];FromDigits[p]];Table[fa[n], {n, 100}] (* Ray Chandler, Oct 18 2006 *)

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

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.

Edited by Andrei Zabolotskii, Jul 20 2025