A156703 -id:A156703 - cp's OEIS frontend

A158911 Numbers of the form 2^i*5^j - 1.

0, 1, 3, 4, 7, 9, 15, 19, 24, 31, 39, 49, 63, 79, 99, 124, 127, 159, 199, 249, 255, 319, 399, 499, 511, 624, 639, 799, 999, 1023, 1249, 1279, 1599, 1999, 2047, 2499, 2559, 3124, 3199, 3999, 4095, 4999, 5119, 6249, 6399, 7999, 8191, 9999, 10239

Offset: 1

Ctibor O. Zizka, Mar 30 2009

Numbers n such that 10^n is divisible by n+1.
Numbers n such that the prime divisors of n+1 are also divisors of the numbers m obtained by the concatenation of n and n+1. For example, for n=39, m = 3940, the divisors of 40 are {2, 5} and the divisors of 3940 are {2, 5, 197}. - Michel Lagneau, Dec 20 2011
The entries correspond to positional information of A156703, which stem from ratios of consecutive integers. For example, A156703(4)=875 yields a(5). This is because 875 was produced from n/(n+1) where n=7, i.e., 7/8 = 0.875.  Similarly, a(23)=399 stems from 399/400=0.9975 (A156703(22)). - Bill McEachen, Jan 05 2014

Robert Israel, Table of n, a(n) for n = 1..10000 (first 134 terms from Peter Pein)

Cf. A158520, A034887, A129344.

[n: n in [0..10^5] | Modexp(10, n, n+1) eq 0]; // Vincenzo Librandi, Mar 07 2018

N:= 20000: # to get all terms <= N
sort([seq(seq(2^i*5^j-1, j=0..floor(log[5]((N+1)/2^i))),i=0..ilog2(N+1))]); # Robert Israel, Mar 06 2018

fQ[n_] := PowerMod[10, n, n + 1] == 0; Select[ Range[0, 11000], fQ] (* Robert G. Wilson v, Sep 08 2010 *)

is(n)=n=(n+1)>>valuation(n+1,2);ispower(n,,&n);n==1||n==5 \\ Charles R Greathouse IV, Jan 12 2012

list(lim)=my(v=List(), N); lim++; for(n=0, log(lim)\log(5), N=5^n; while(N<=lim, listput(v, N-1); N<<=1)); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jan 12 2012

a(n) = A003592(n) - 1.

Corrected by Claudio Meller, Rick L. Shepherd, Charles R Greathouse IV and R. J. Mathar, Aug 23 2010

Edited by N. J. A. Sloane, Aug 25 2010, Oct 04 2010

A259299 The decimal expansion of n/(n+1) until it terminates or repeats, shown without the decimal point.

Original entry on oeis.org

0, 5, 6, 75, 8, 83, 857142, 875, 8, 9, 90, 916, 923076, 9285714, 93, 9375, 9411764705882352, 94, 947368421052631578, 95, 952380, 954, 9565217391304347826086, 9583, 96, 9615384, 962, 96428571, 9655172413793103448275862068, 96, 967741935483870, 96875, 96, 97058823529411764, 9714285, 972, 972

Offset: 0

Doug Bell, Jun 23 2015

The first occurrence of a repeated term where a(n) = a(n+1) is for a(35) and a(36), both of which equal 972.  This results from two different repeating decimals with different length repeating periods but the same non-repeating plus repeating digits, namely 35/36 = .972222... = 972 (repeating period of 1) and 36/37 = .972972... = 972 (repeating period of 3).
Other than n = (36,37) the only repeated terms appear to follow one of the following two patterns for the larger value of n:
  First pattern: for n >= 111, where all digits of n are 1: 111, 1111, 11111, ... and a(n-1) = a(n) = 990, 9990, 99990, ... with the repeating decimal for ((n-1)/n, n/(n+1)) of (.9909090..., .990990990...), (.9990990990..., .999099909990...), (.9999099909990..., .999909999099990...).  Where d is the number of digits in a(n), the repeating period for the decimal values is (d-1, d).
  Second pattern: for n >= 10101, where the digits of n alternate between 0 and 1, with a final digit of 1: 10101, 1010101, 101010101, ... and a(n-1) = a(n) = 999900, 9999900, 99999900, ... with the repeating decimal for ((n-1)/n, n/(n+1)) of (.99990099009900..., .999900999900999900...), (.99999009990099900..., .999990099999009999900...), (.99999900999900999900..., .999999009999990099999900...).   Where d is the number of digits in a(n), the repeating period for the decimal values is (d-2, d).
Have verified that there are no other repeating terms up to n = 10^6.

			a(1)=5 (1/2=0.5), a(2)=6 (2/3=0.6666...=6), a(3)=75 (3/4=0.75=75).

Doug Bell, Table of n, a(n) for n = 0..1000

Subsequences A156703, A235589.

Cf. A036275, A060284.

Array[FromDigits@ Flatten@ First@ RealDigits[(# - 1)/#] &, 37] (* Michael De Vlieger, Aug 18 2015 *)

A235589 The periodic part of the decimal expansion of m/(m+1), for those m/(m+1) that have pure periods.

Original entry on oeis.org

6, 857142, 8, 90, 923076, 9411764705882352, 947368421052631578, 952380, 9565217391304347826086, 962, 9655172413793103448275862068, 967741935483870, 96, 972, 974358, 97560, 976744186046511627906, 9787234042553191489361702127659574468085106382, 979591836734693877551020408163265306122448

Offset: 1

Bill McEachen, Jan 12 2014

A companion sequence stemming from the some of the elements excluded by A156703. The sequence is highly volatile and infinite...as with A156703 the subset elements are encountered in numerical order. a(n) will start with the digit 9 for n>4 I believe. Entries can grow quite large very quickly. Each entry will be encountered once, and they will end in an even digit.

The number of digits of a(n) is given by A002329. - Michel Marcus, Aug 19 2015

			1/2=0.5 non-repeating, so exclude from sequence.
2/3=0.6 repeating, so a(1)=6.
5/6=0.833 (repeating) but has "8" prefix ahead of repeating "3" so exclude from sequence (decimal expansion not purely periodic)
6/7=0.857142 repeating so a(2)=857142.

Subsequence of A259299.

Cf. A156703, A036275, A060283, A114205.

FromDigits@ Flatten@ First@ RealDigits[(# - 1)/#] & /@ Select[Range@ 120, CoprimeQ[#, 10] &] //Rest (* Michael De Vlieger, Aug 18 2015 *)

a(n) = the periodic part of the decimal expansion of (A045572(n+1)-1) / A045572(n+1). - Doug Bell, Aug 17 2015

Missing terms added by Ralf Stephan, Jan 19 2014

Incorrect terms 916, 94 removed and two more terms added by Michael De Vlieger, Aug 18 2015

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A158911 Numbers of the form 2^i*5^j - 1.

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A259299 The decimal expansion of n/(n+1) until it terminates or repeats, shown without the decimal point.

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Mathematica

A235589 The periodic part of the decimal expansion of m/(m+1), for those m/(m+1) that have pure periods.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

Extensions