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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

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

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

EXAMPLE

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.

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 *)

PROG

(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 n<maxx:

...q=Decimal(n)/Decimal(n+1)

...ratio=str(q)

...myLen=len(ratio)

...ratio.replace(" ", "")

...if len(ratio[2:])<15:

......print (ratio[2:])

...else:

......strCopy=copy.copy(ratio[2:])

......match=0

......maxCnt=0

......keyStr=' '

......subLen=n

......cap=len(ratio[2:])

......for j5 in range(0, cap ):

.........for i5 in range(subLen, 1, -1):

............if i5<=j5:

...............break

............subStr=strCopy[j5:i5]

............if len(subStr)<1:

...............continue

............match=strCopy.count(subStr)

............z=match*len(subStr)

............if z>maxCnt and match>1:

...............if len(subStr)==1 and z<subLen:

..................maxCnt=z

..................keyStr=copy.copy(subStr)

...............else:

..................maxCnt=z

..................keyStr=copy.copy(subStr)

......if maxCnt>4:

.........pass

......else:

.........print (ratio[2:] )

......getcontext().prec = max(2*subLen, 200)

...n+=1

# Bill McEachen, Dec 28 2015

CROSSREFS

Cf. A003592, A158911. See comment at A158911.

Sequence in context: A192564 A080473 A266570 * A285452 A048350 A030991

Adjacent sequences:  A156700 A156701 A156702 * A156704 A156705 A156706

KEYWORD

easy,nonn,base

AUTHOR

Bill McEachen, Feb 13 2009

STATUS

approved

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Last modified September 20 05:51 EDT 2019. Contains 327212 sequences. (Running on oeis4.)