A002371 -id:A002371 - cp's OEIS frontend

A284602 Numbers k such that the decimal representation of 1/k is either finite or has even period.

1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 16, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 32, 33, 34, 35, 38, 39, 40, 42, 44, 46, 47, 49, 50, 51, 52, 55, 56, 57, 58, 59, 61, 63, 64, 65, 66, 68, 69, 70, 73, 76, 77, 78, 80, 84, 85, 87, 88, 89, 91, 92, 94, 95, 97, 98, 99, 100, 101, 102, 103, 104, 105, 109, 110, 112, 113, 114, 115

Offset: 1

Ilya Gutkovskiy, Mar 30 2017

All numbers of the form 2^i*5^j with i, j >= 0 are in this sequence (numbers with a finite decimal expansion).
From Robert G. Wilson v, Apr 02 2017: (Start)
If k is in the sequence, then so are 2k and 5k.
The complement of A284601.
Primitives: 1, 7, 11, 13, 17, 19, 21, 23, 29, 33, 39, 47, 49, 51, 57, 59, 61, 63, ..., .
(End)

			14 is in the sequence because 1/14 = 0.0714285(714285)..., whose period is 6, an even number.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Index entries for sequences related to decimal expansion of 1/n

Cf. A002371, A003592, A028416, A051626, A206586, A284601.

Select[Range[115], Mod[Length[RealDigits[1/#][[1, -1]]], 2] == 0 & ]

A122060 Position in decimal expansion of 1/n where repetition begins.

Original entry on oeis.org

2, 3, 2, 4, 3, 3, 7, 5, 2, 3, 3, 4, 7, 8, 3, 6, 17, 3, 19, 4, 7, 4, 23, 5, 4, 8, 4, 11, 29, 3

Offset: 1

Ben Paul Thurston, Sep 14 2006

If 1/n = 0.XYYYYY... then sequence gives index of first digit of the second Y.

a(4) = 4 and a(p) = p for primes p = {7, 17, 19, 23, 29, 47, 59, 61, 97, ...} = A001913(n) Cyclic numbers: primes with primitive root 10. - Alexander Adamchuk, Jan 28 2007

			a(4) = 4 because in 0.2500 the zero begins repeating at the fourth position.
a(17) = 17 because 0.05882352941176470588... begins repeating at the 17th position.

Cf. A001913, A002371.

a(n)=A121341(n)+2 if 1/n terminates, else a(n)=A121341(n)+1. - R. J. Mathar, Sep 20 2006

A122183 Primes p_i by index i for which the period length of 1/p_i is a semiprime.

Original entry on oeis.org

4, 6, 9, 11, 14, 15, 17, 19, 20, 26, 27, 34, 39, 41, 43, 56, 59, 61, 62, 64, 76, 83, 85, 86, 96, 101, 102, 109, 111, 112, 119, 124, 138, 140, 144, 147, 149, 150, 154, 161, 166, 168, 170, 171, 175, 192, 198, 203, 216, 219, 222, 224, 225, 235, 236, 239, 240, 246, 251

Offset: 1

Jonathan Vos Post, May 10 2007

Semiprime analog of A072859 based on A002371.

Numbers n such that A002371(n) is an element of A001358.

			a(1) = 4 because A002371(4) Period of decimal expansion of 1/(4th prime) = 6 = 2*3, a semiprime.
a(2) = 6 because A002371(6) = 6 = 2*3.
a(3) = 9 because A002371(9) = 22 = 2*11.
a(4) = 11 because A002371(11) = 15 = 3*5.
a(5) = 14 because A002371(14) = 21 = 3*7.
a(6) = 15 because A002371(15) = 46 = 2*23.
a(7) = 17 because A002371(17) = 58 = 2*29.

Cf. A000040, A001358, A002371, A072859, A128948.

semiprimeQ[n_] := Plus @@ Last /@ FactorInteger[n] == 2; PrimePi /@ Select[Prime@ Range@ 254, semiprimeQ@ MultiplicativeOrder[10, # ] &] (* Robert G. Wilson v *)

Edited, corrected and extended by Robert G. Wilson v, May 22 2007

A212528 The periodic part of the decimal expansion of prime(n-1) / prime(n).

Original entry on oeis.org

6, 0, 714285, 63, 846153, 7647058823529411, 894736842105263157, 8260869565217391304347, 7931034482758620689655172413, 935483870967741, 837, 90243, 953488372093023255813, 9148936170212765957446808510638297872340425531, 8867924528301

Offset: 2

Jaroslav Krizek, Jun 29 2012

Number of digits of the periodic parts for n>=3 in A002371, A048595 and A071126.

Jaroslav Krizek, Table of n, a(n) for n = 2..60

Cf. A002371, A048595, A071126.

Table[If[n == 3, 0, FromDigits[RealDigits[Prime[n-1]/Prime[n]][[1,1]]]], {n, 2, 10}]

Corrected by T. D. Noe, Jun 29 2012

A227969 Powers of primes other than 2 and 5 in order by cycle length of reciprocal in decimal.

Original entry on oeis.org

3, 9, 11, 27, 37, 101, 41, 271, 7, 13, 239, 4649, 73, 137, 81, 333667, 9091, 21649, 513239, 9901, 53, 79, 265371653, 909091, 31, 2906161, 17, 5882353, 2071723, 5363222357, 19, 52579, 1111111111111111111, 3541, 27961, 43, 1933, 10838689, 23, 121, 4093, 8779, 11111111111111111111111

Offset: 1

J. Lowell, Aug 01 2013

			3 and 9 qualify for the first 2 terms because both of them have a reciprocal cycle of 1. Then 11 has a reciprocal cycle of 2; then 27 and 37 have 3; then 101 has 4; then 41 and 271 have 5. Table begins:
3, 9;
11;
27, 37;
101;
41, 271;
7, 13;
239, 4649;
73, 137;
81, 333667;
9091;
21649, 513239;
9901;
53, 79, 265371653;

Charles R Greathouse IV, Rows n = 1..100 of triangle, flattened

Cf. A002371, A046107.

go(n)=my(v=[],P=[],E=[],t,ok); for(k=1,n, t=setminus(factor(10^k-1)[,1]~,P); E=concat(E,vector(#t,i,1)); P=concat(P,t); E=apply(i->E[i],Vec(vecsort(P,,1))); P=vecsort(P); ok=1; while(ok, ok=0; for(i=1,#P,if(znorder(Mod(10,P[i]^(E[i]+1)))==k, E[i]++; t=concat(t,P[i]^E[i]); ok=1))); v=concat(v,t=vecsort(t)); print(k" "t)); v \\ Charles R Greathouse IV, Aug 01 2013

a(9)-a(43) from Charles R Greathouse IV, Aug 01 2013

A291943 a(0)=0; for n>0, a(n) = (2n)-th digit after the decimal point in the decimal expansion of 1/(2n+1).

Original entry on oeis.org

0, 3, 0, 7, 1, 9, 3, 6, 7, 1, 4, 3, 0, 3, 1, 9, 3, 5, 7, 2, 9, 3, 2, 7, 8, 1, 3, 1, 1, 1, 9, 1, 3, 7, 1, 9, 3, 3, 9, 1, 7, 3, 7, 1, 1, 9, 1, 5, 7, 1, 9, 3, 0, 7, 1, 0, 3, 6, 0, 0, 8, 0, 0, 7, 0, 9, 8, 0, 7, 1, 0, 9, 8, 4, 1, 9, 4, 4, 7, 0, 6, 3, 0, 7, 3, 5, 3, 4, 0, 1, 9, 0, 4, 5, 0, 9, 3, 0, 7, 1

Offset: 0

Marco Matosic, Sep 06 2017

			a(3)=7 since we want the sixth decimal digit of 1/7.

John H. Conway & Richard K. Guy, The Book of Numbers; Springer 1996.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A002371, A003060, A006883, A036275, A051626, A061480, A114205, A121090, A270392.

f:= proc(n) floor(10^(2*n)/(2*n+1)) mod 10 end proc:
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Oct 31 2017

f[n_] := Mod[Floor[10^(2n)/(2n +1)], 10]; f[0] = 0; Array[f, 105, 0] (* Robert G. Wilson v, Oct 31 2017 *)

Edited by N. J. A. Sloane, Oct 30 2017

a(82) corrected by Robert Israel, Oct 31 2017

A060263 a(n) = smallest prime q such that precisely n successive primes p starting at q have reciprocals with period p-1 and prevprime(q) is not such a prime.

Original entry on oeis.org

59, 257, 17, 487, 5737, 23459, 364379, 681899, 4275343, 14747137, 12284017, 61598897, 62232899, 95386019, 824443051, 2245849783

Offset: 2

Jeff Burch, Mar 23 2001

Cf. A002371, A048595, A060262.

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 21 2003

a(6) onward corrected and title clarified by Sean A. Irvine, Nov 05 2022

A172372 Least number k such that the n-th prime not dividing 10 (A004139(n)) divides the repunit (10^k-1)/9.

Original entry on oeis.org

3, 6, 2, 6, 16, 18, 22, 28, 15, 3, 5, 21, 46, 13, 58, 60, 33, 35, 8, 13, 41, 44, 96, 4, 34, 53, 108, 112, 42, 130, 8, 46, 148, 75, 78, 81, 166, 43, 178, 180, 95, 192, 98, 99, 30, 222, 113, 228, 232, 7, 30, 50, 256, 262, 268, 5, 69, 28, 141, 146, 153, 155, 312, 79, 110

Offset: 1

Michel Lagneau, Feb 01 2010

nonn

If p is an odd prime different from 5, then p divides an infinite number of terms of the sequence of repunits {1, 11, 111, 1111, ... }. The proof is elementary: let p be such a prime. If p = 3, then 3 divides (10^3-1)/9 = 111. Otherwise, take k = (10^p - 1)/9; by the Fermat theorem, 10^(p-1) == 1 (mod p), so p divides (10^(p-1)-1); since p is relatively prime to 9, it divides k. Trivially, if p divides any k digit repunit, it divides the k*m digit repunit as well.

Essentially the same as A002371. - T. D. Noe, Apr 11 2012

			3 divides 111, but not 1 or 11, so a(1) = 3.
7 divides 111111 but not 1, 11, 111, 1111, or 11111, so a(2) = 6.

David Wells, The Factors of the Repunits 11 through R_40, The Penguin Dictionary of Curious and Interesting Numbers, p. 219 Penguin 1986.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, 1997.
David Wells, Curious and Interesting Numbers (Revised), Penguin Books, page 114.

Project Euler, Problem 129: Repunit divisibility
S. S. Wagstaff, Jr., The Cunningham Project
Eric Weisstein's World of Mathematics, Repunit.

Cf. A002275 (repunits), A002371 (period of decimal expansion of 1/prime(n)), A004139 (odd primes excluding 5), A095250 (11111111... (n times) mod n).

a(n) = {k=1; p = if(n>1, prime(n+2), 3); while((10^k-1)/9 % p, k++); k;} \\ Michel Marcus, May 25 2014

Corrected and edited by Michel Lagneau, Apr 25 2010

Term 6 between terms 44 and 96 doesn't belong to the sequence. The same for term 43 between terms 43 and 178. Corrected and edited by Krzysztof Wojtas, May 25 2014

A247585 Period of the decimal expansion of 1/p as p runs through the prime numbers of the form n^2+1 (0 by convention for the primes 2 and 5).

Original entry on oeis.org

0, 0, 16, 3, 4, 98, 256, 200, 576, 338, 1296, 200, 1458, 3136, 242, 1369, 7056, 1620, 4418, 12100, 13456, 3600, 15376, 567, 3380, 8978, 10658, 7500, 24336, 25, 5780, 30976, 600, 33856, 41616, 10609, 44100, 50176, 52900, 55696, 14400, 62500, 65536, 33800, 69696, 8100

Offset: 1

Michel Lagneau, Sep 20 2014

nonn

Subsequence of A002371 or period of the decimal expansion of 1/A002496(n).

The squares > 0 in the sequence are 4, 16, 25, 256, 576, 1296, 1369, 3136, 3600, 7056, 8100, 10609, ...

			a(3) = 16 because A002496(3) = 17 and 1/17 = 0. 0588235294117647 0588235294117647 ... has period 16.

Cf. A000720, A002371, A002496, A060282, A175550.

lst={};Do[If[PrimeQ[n^2+1],AppendTo[lst,n^2+1]],{n,1,1000}];Table[Length[RealDigits[1/lst[[m]]][[1,1]]],{m,1,60}]

a(n) = A002371(A000720(A002496(n))). [Corrected by Georg Fischer, Oct 19 2024]

A338792 a(n) is the least number k such that 1/prime(k) has repeating decimal expansion of period n.

Original entry on oeis.org

1, 2, 5, 12, 26, 13, 4, 52, 21, 28693, 1128, 2431, 1221, 16, 71954, 11, 7, 153888, 8, 27417323062119920, 496, 14, 9, 223378173194137397198, 5760923, 2403, 149, 134, 10, 452, 47, 406, 71, 19, 27, 20, 37607875619, 150886, 22544062111497849

Offset: 0

Ilya Gutkovskiy, Nov 09 2020

			1/prime(1)  = 1/2  = 0.5 (finite decimal expansion).
1/prime(2)  = 1/3  = 0.3(3)... (period 1).
1/prime(5)  = 1/11 = 0.09(09)... (period 2).
1/prime(12) = 1/37 = 0.027(027)... (period 3).

Index entries for sequences related to decimal expansion of 1/n

Cf. A000720, A002371, A003060, A007138.

a(0) = 1; a(n) = A000720(A007138(n)).

a(19)-a(38) from Daniel Suteu, Nov 09 2020 [using data from A007138 and A234317]

cp's OEIS Frontend

A284602 Numbers k such that the decimal representation of 1/k is either finite or has even period.

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A122060 Position in decimal expansion of 1/n where repetition begins.

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A122183 Primes p_i by index i for which the period length of 1/p_i is a semiprime.

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A212528 The periodic part of the decimal expansion of prime(n-1) / prime(n).

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A227969 Powers of primes other than 2 and 5 in order by cycle length of reciprocal in decimal.

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A291943 a(0)=0; for n>0, a(n) = (2n)-th digit after the decimal point in the decimal expansion of 1/(2n+1).

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A060263 a(n) = smallest prime q such that precisely n successive primes p starting at q have reciprocals with period p-1 and prevprime(q) is not such a prime.

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A172372 Least number k such that the n-th prime not dividing 10 (A004139(n)) divides the repunit (10^k-1)/9.

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A247585 Period of the decimal expansion of 1/p as p runs through the prime numbers of the form n^2+1 (0 by convention for the primes 2 and 5).