A077313
Primes of the form 2^r*5^s - 1.
Original entry on oeis.org
3, 7, 19, 31, 79, 127, 199, 499, 1249, 1279, 1999, 4999, 5119, 8191, 12799, 20479, 31249, 49999, 51199, 79999, 81919, 131071, 199999, 524287, 799999, 1249999, 1310719, 3124999, 3276799, 4999999, 7812499, 12499999, 19999999, 20479999
Offset: 1
1250000 = 2*2*2*2*5*5*5*5*5*5*5 and 1250000 - 1 = A000040(96469), therefore 1249999 is a term.
List of (r, s): (2, 0), (3, 0), (2, 1), (5, 0), (4, 1), (7, 0), (3, 2), (2, 3), (1, 4), (8, 1), (4, 3), (3, 4), (10, 1), ... - _Muniru A Asiru_, Sep 29 2017
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A:=Filtered([1..10^7],IsPrime);; I:=[5];;
B:=List(A,i->Elements(Factors(i+1)));;
C:=List([0..Length(I)],j->List(Combinations(I,j),i->Concatenation([2],i)));;
A077313:=List(Set(Flat(List([1..Length(C)],i->List([1..Length(C[i])],j->Positions(B,C[i][j]))))),i->A[i]); # Muniru A Asiru, Sep 29 2017
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With[{n = 10^8}, Union@ Select[Flatten@ Table[2^p*5^q - 1, {p, 0, Log[2, n/(1)]}, {q, 0, Log[5, n/(2^p)]}], PrimeQ]] (* Michael De Vlieger, Sep 30 2017 *)
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.
Original entry on oeis.org
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
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.
- 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).
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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
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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 *)
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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
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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
A212720
The periodic part of the decimal expansion of n/(n+1). Any initial 0's are to be placed at end of cycle.
Original entry on oeis.org
0, 6, 0, 0, 3, 857142, 0, 8, 0, 90, 6, 923076, 285714, 3, 0, 9411764705882352, 4, 947368421052631578, 0, 952380, 54, 9565217391304347826086, 3, 0, 615384, 962, 428571, 9655172413793103448275862068, 6, 967741935483870, 0, 96, 7058823529411764, 714285, 2, 972
Offset: 1
6/7 = 0.85714285714285714285714285714286... and digit-cycle is 857142, so a(6) = 857142.
Cf.
A036275 (periodic part of the decimal expansion of 1/n),
A158911 (numbers n such that a(n)=0),
A051037.
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Table[FromDigits[FindTransientRepeat[RealDigits[n/(n+1),10,100][[1]],3][[2]]],{n,40}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 07 2017 *)
Showing 1-3 of 3 results.
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