A156703 String of digits encountered in decimal expansion of successive ratios k/(k+1), treating only non-repeating expansions, with decimal point and leading and trailing zeros removed.

5, 75, 8, 875, 9, 9375, 95, 96, 96875, 975, 98, 984375, 9875, 99, 992, 9921875, 99375, 995, 996, 99609375, 996875, 9975, 998, 998046875, 9984, 9984375, 99875, 999, 9990234375, 9992, 99921875, 999375, 9995, 99951171875, 9996, 999609375, 99968, 9996875, 99975

Offset: 1

Author

Bill McEachen, Feb 13 2009

Keywords

• easy
• nonn
• base

Comments

The sequence seems infinite and may be volatile in its extrema.

Conjecture: subsets of the sequence (as it fills out) will correspond to the odd integers by length.

Thus, there are 3 single-digit entries in range {1-9}, ending at 9; 5 two-digit entries in range {10-99} ending at 99; 7 three-digit entries in range {100-999} ending at 999, etc. The remainder set of course are all repeating decimals.

Denominators of the ratios that yield each term must be terms of A003592 (i.e., any integer m whose distinct prime factors p also divide 10, or m regular to 10), since only these denominators produce non-repeating decimal expansions. - Michael De Vlieger, Dec 30 2015

Examples

			1/2 = 0.5 (non-repeating), which yields a(1) = 5.
2/3 = 0.6666... (repeating, so does not yield a term in the sequence).
3/4 = 0.75 (non-repeating), which yields a(2) = 75.
4/5 = 0.8 (non-repeating), which yields a(3) = 8.
		

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Sixth Edition, Oxford University Press, 2008, pages 141-144 (including Theorem 135).

Links

• Robert Israel, Table of n, a(n) for n = 1..6000
• Eric Weisstein's World of Mathematics, Decimal Expansion.
• Eric Weisstein's World of Mathematics, Regular Number.
• Wikimedia Commons, Alternate plot.

Crossrefs

Cf. A003592, A158911. See comment at A158911.

Programs

Maple

N:= 10^5: # to get terms for denominators <= N
B:= sort([seq(seq(2^i*5^j,i=0..ilog2(N/5^j)),j=0..ilog(N,5))]):
seq(10^max(padic:-ordp(n,2),padic:-ordp(n,5))*(n-1)/n, n=B[2..-1]); # Robert Israel, Dec 29 2015

Mathematica

FromDigits@ First@ # & /@ RealDigits@ Apply[#1/#2 &, Transpose@ {# - 1, #} &@ Select[Range@ 10000, AllTrue[First /@ FactorInteger@ #, MemberQ[{2, 5}, #] &] &], 1] (* Michael De Vlieger, Dec 30 2015, Version 10 *)
FromDigits@ First@# & /@ RealDigits@ Apply[#1/#2 &, Transpose@ {# - 1, #} &@ Select[Range@ 10000, First@ Union@ Map[MemberQ[{2, 5}, #] &, First /@ FactorInteger@ #] &], 1] (* Michael De Vlieger, Dec 30 2015, Version 6 *)

PARI

list(maxx)={my(N, vf=List()); maxx++;for(n=0, log(maxx)\log(5),
N=5^n; maxVal= 0;while(N<=maxx, if (N != 1, listput(vf, (N-1)/N));
N<<=1;)); vf = vecsort(Vec(vf));for (i=1,length(vf),
while(denominator(vf[i]) != 1, vf[i] *= 10););print(vf);}
\\ adapted from A158911 code, courtesy Michel Marcus, Dec 29 2015

Python

import string,copy
from decimal import *
getcontext().prec = 200
maxx=1000
n=1
maxLen=0
while nmaxCnt and match>1:
                    if len(subStr)==1 and z4:
            pass
        else:
            print(ratio[2:])
        getcontext().prec = max(2*subLen,200)
    n+=1
# Bill McEachen, Dec 28 2015

Formula

a(n) = 10^d*(k-1)/k where k = A003592(n+1) = 2^i*5^j and d=max(i,j). - Robert Israel, Dec 29 2015