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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 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 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 June 21 06:55 EDT 2021. Contains 345358 sequences. (Running on oeis4.)