A048595 -id:A048595 - cp's OEIS frontend

A002371 Period of decimal expansion of 1/(n-th prime) (0 by convention for the primes 2 and 5).

0, 1, 0, 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

N. J. A. Sloane

a(n) is the minimum solution x of modular equation 10^x == 1 (mod p), where p = prime(n). - Carmine Suriano, Oct 10 2012
a(n) = smallest m such that 111...11 (m 1's) is divisible by the n-th prime, or 0 if no such m exists (with the exception that a(2) = 3 instead of 1). E.g., the 5th prime, 11, divides 11, so a(5) = 2. - N. J. A. Sloane, Oct 03 2013 [Comment corrected by Derek Orr, Jun 14 2014]
Numbers n such that A071126(n) = A000040(n) - 1. - Hugo Pfoertner, Mar 18 2003
Except for n = 1 and 3, a(n) divides A006093(n). - Robert Israel, Jul 15 2016

			A002371(11) = 15 because the 11th prime is 31, and 1/31 = 0.03225806451612903225806451612903225806452... has period 15. - _Richard F. Lyon_, Mar 29 2022

Albert H. Beiler, Recreations in the Theory of Numbers, 2nd ed. New York: Dover, 1966, pages 65, 309. ISBN 0-486-21096-0.
John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, 1996, p. 162. ISBN 978-0-387-97993-9.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 15.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
C. K. Caldwell, The Prime Glossary, Period of a prime
Matt Parker and Brady Haran, The Reciprocals of Primes, Numberphile video (2022).
William Shanks, On the number of figures in the period of the reciprocal of every prime number below 20,000, Proc. Royal Soc. London, 22 (1874), 200-210. See also on JSTOR.
Eric Weisstein's World of Mathematics, Decimal Expansion
Index entries for sequences related to decimal expansion of 1/n

See A048595 for another version. Cf. A006883, A007732, A051626, A071126, A000040, A002275, A097443.
Cf. A001913 (full repetend primes), A060257 (1/prime(n) has period prime(n) - 1).
In other bases: A014664, A062117, A211241, A211242, A211243, A211244, A211245, A002371.

seq(subs(FAIL=0,numtheory:-order(10, ithprime(n))),n=1..100); # Robert Israel, Jul 15 2016

Table[ Length[ RealDigits[1 / Prime[n]] [[1, 1]]], {n, 1, 70}]
Table[If[IntegerQ[#], #, 0] &[MultiplicativeOrder[10, Prime[n]]], {n, 1, 70}] (* Jan Mangaldan, Jul 07 2020 *)

a(n)=if(n<4,n==2,znorder(Mod(10, prime(n))))

from sympy import prime, n_order
def A002371(n): return 0 if n == 1 or n == 3 else n_order(10,prime(n)) # Chai Wah Wu, Feb 07 2022

From Alexander Adamchuk, Jan 28 2007: (Start)
a(A000720(p)) = p - 1 for primes p in A001913.
a(A060257(n)) = prime(A060257(n)) - 1. (End)

More terms from Arlin Anderson (starship1(AT)gmail.com)

Edited by Charles R Greathouse IV, Mar 24 2010

A051626 Period of decimal representation of 1/n, or 0 if 1/n terminates.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 6, 0, 1, 0, 2, 1, 6, 6, 1, 0, 16, 1, 18, 0, 6, 2, 22, 1, 0, 6, 3, 6, 28, 1, 15, 0, 2, 16, 6, 1, 3, 18, 6, 0, 5, 6, 21, 2, 1, 22, 46, 1, 42, 0, 16, 6, 13, 3, 2, 6, 18, 28, 58, 1, 60, 15, 6, 0, 6, 2, 33, 16, 22, 6, 35, 1, 8, 3, 1, 18, 6, 6, 13, 0, 9, 5, 41, 6, 16, 21, 28, 2, 44, 1

Offset: 1

J. Lowell

Essentially same as A007732.

For any prime number p: if a(p) > 0, a(p) divides p-1. - David Spitzer, Jan 09 2017

			From _M. F. Hasler_, Dec 14 2015: (Start)
a(1) = a(2) = 0 because 1/1 = 1 and 1/2 = 0.5 have a finite decimal expansion.
a(3) = a(6) = a(9) = a(12) = 1 because 1/3 = 0.{3}*, 1/6 = 0.1{6}*, 1/9 = 0.{1}*, 1/12 = 0.08{3}* where the sequence of digits {...}* which repeats indefinitely is of length 1.
a(7) = 6 because 1/7 = 0.{142857}* with a period of 6.
a(17) = 16 because 1/17 = 0.{0588235294117647}* with a period of 16.
a(19) = 18 because 1/19 = 0.{052631578947368421}* with a period of 18. (End)

T. D. Noe, Table of n, a(n) for n = 1..10000
Project Euler, Reciprocal cycles: Problem 26
Eric Weisstein's World of Mathematics, Repeating Decimal
Index entries for sequences related to decimal expansion of 1/n

Essentially same as A007732. Cf. A002371, A048595, A006883, A036275, A114205, A114206, A001913.

A051626 := proc(n) local lpow,mpow ;
    if isA003592(n) then
       RETURN(0) ;
    else
       lpow:=1 ;
       while true do
          for mpow from lpow-1 to 0 by -1 do
              if (10^lpow-10^mpow) mod n =0 then
                 RETURN(lpow-mpow) ;
              fi ;
          od ;
          lpow := lpow+1 ;
       od ;
    fi ;
end: # R. J. Mathar, Oct 19 2006

r[x_]:=RealDigits[1/x]; w[x_]:=First[r[x]]; f[x_]:=First[w[x]]; l[x_]:=Last[w[x]]; z[x_]:=Last[r[x]];
d[x_] := Which[IntegerQ[l[x]], 0, IntegerQ[f[x]]==False, Length[f[x]], True, Length[l[x]]]; Table[d[i], {i,1,90}] (* Hans Havermann, Oct 19 2006 *)
fd[n_] := Block[{q},q = Last[First[RealDigits[1/n]]];If[IntegerQ[q], q = {}]; Length[q]];Table[fd[n], {n, 100}] (* Ray Chandler, Dec 06 2006 *)
Table[Length[RealDigits[1/n][[1,-1]]],{n,90}] (* Harvey P. Dale, Jul 03 2011 *)
a[n_] := If[ PowerMod[10, n, n] == 0, 0, MultiplicativeOrder[10, n/2^IntegerExponent[n, 2]/5^IntegerExponent[n, 5]]]; Array[a, 90] (* myself in A003592 and T. D. Noe in A007732 *) (* Robert G. Wilson v, Feb 20 2025 *)

A051626(n)=if(1M. F. Hasler, Dec 14 2015

def A051626(n):
    if isA003592(n):
        return 0
    else:
        lpow=1
        while True:
            for mpow in range(lpow-1,-1,-1):
                if (10**lpow-10**mpow) % n == 0:
                    return lpow-mpow
            lpow += 1 # Kenneth Myers, May 06 2016

from sympy import multiplicity, n_order
def A051626(n): return 0 if (m:=(n>>(~n & n-1).bit_length())//5**multiplicity(5,n)) == 1 else n_order(10,m) # Chai Wah Wu, Aug 11 2022

a(n)=A132726(n,1); a(n)=a(A132740(n)); a(A132741(n))=a(A003592(n))=0. - Reinhard Zumkeller, Aug 27 2007

More terms from James Sellers

A006556 Number of different cycles of digits in the decimal expansions of 1/p, 2/p, ..., (p-1)/p where p = n-th prime different from 2 or 5.

Original entry on oeis.org

2, 1, 5, 2, 1, 1, 1, 1, 2, 12, 8, 2, 1, 4, 1, 1, 2, 2, 9, 6, 2, 2, 1, 25, 3, 2, 1, 1, 3, 1, 17, 3, 1, 2, 2, 2, 1, 4, 1, 1, 2, 1, 2, 2, 7, 1, 2, 1, 1, 34, 8, 5, 1, 1, 1, 54, 4, 10, 2, 2, 2, 2, 1, 4, 3, 1, 2, 3, 11, 2, 1, 2, 1, 1, 1, 4, 2, 2, 1, 3, 2, 1, 2, 2, 14, 3, 1, 3, 2, 2, 1, 1, 1, 1, 1, 10, 2, 1, 6

Offset: 3

N. J. A. Sloane

			1/13=.0769230769..., 2/13=.1538461538..., 3/13= .2307692307..., etc., with 2 different cycles, so a(4) = 2 [13 is the 4th prime different from 2 or 5].

J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 162.
M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 131.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=3..1000
Victor Meally, Letter to N. J. A. Sloane, no date.

See A048595 and A002371 for the length of the cycles. See also A054471.

Map[(# - 1)/MultiplicativeOrder[10, #] &, {3}~Join~Prime@ Range[4, 101]] (* Michael De Vlieger, May 27 2020 *)

f(p) = (p-1)/znorder(Mod(10, p));
lista(nn) = {my(vp=select(x->(10%x), primes(nn))); apply(f, vp);} \\ Michel Marcus, May 27 2020

(p-1)/x, where 10^x = 1 mod p.

More terms from James Sellers, May 24 2000

Edited by Charles R Greathouse IV, Nov 01 2009

A036275 The periodic part of the decimal expansion of 1/n. Any initial 0's are to be placed at end of cycle.

Original entry on oeis.org

0, 0, 3, 0, 0, 6, 142857, 0, 1, 0, 90, 3, 769230, 714285, 6, 0, 5882352941176470, 5, 526315789473684210, 0, 476190, 45, 4347826086956521739130, 6, 0, 384615, 370, 571428, 3448275862068965517241379310, 3, 322580645161290, 0, 30, 2941176470588235, 285714, 7

Offset: 1

Floor van Lamoen

a(n) = 0 iff n = 2^i*5^j (A003592). - Jon Perry, Nov 19 2014

a(n) = n iff n = 3 or 6 (see De Koninck & Mercier reference). - Bernard Schott, Dec 02 2020

			1/28 = .03571428571428571428571428571428571428571... and digit-cycle is 571428, so a(28)=571428.

Jean-Marie De Koninck & Armel Mercier, 1001 Problèmes en Théorie Classique des Nombres, Problème 347 pp. 50 and 205, Ellipses, Paris, 2004.

Philippe Guglielmetti, Table of n, a(n) for n = 1..1001 (first 500 terms from T. D. Noe)
Index entries for sequences related to decimal expansion of 1/n

Cf. A007732, A051626, A002371, A048595, A006883, A007498, A007615, A040017, A051627.
See also A060282, A060283, A060251.
A051628 is length of preamble.

isCycl := proc(n) local ifa,i ; if n <= 2 then RETURN(false) ; fi ; ifa := ifactors(n)[2] ; for i from 1 to nops(ifa) do if op(1,op(i,ifa)) <> 2 and op(1,op(i,ifa)) <> 5 then RETURN(true) ; fi ; od ; RETURN(false) ; end: A036275 := proc(n) local ifa,sh,lpow,mpow,r ; if not isCycl(n) then RETURN(0) ; else lpow:=1 ; while true do for mpow from lpow-1 to 0 by -1 do if (10^lpow-10^mpow) mod n =0 then r := (10^lpow-10^mpow)/n ; r := r mod (10^(lpow-mpow)-1) ; while r*10 < 10^(lpow-mpow) do r := 10*r ; od ; RETURN(r) ; fi ; od ; lpow := lpow+1 ; od ; fi ; end: for n from 1 to 60 do printf("%d %d ",n,A036275(n)) ; od ; # R. J. Mathar, Oct 19 2006

fc[n_]:=Block[{q=RealDigits[1/n][[1,-1]]},If[IntegerQ[q],0,While[First[q]==0,q=RotateLeft[q]];FromDigits[q]]];
Table[fc[n],{n,36}] (* Ray Chandler, Nov 19 2014, corrected Jun 27 2017 *)
Table[FromDigits[FindTransientRepeat[RealDigits[1/n,10,120][[1]],3] [[2]]],{n,40}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Mar 12 2019 *)

Corrected and extended by N. J. A. Sloane

Corrected a(92), a(208), a(248), a(328), a(352) and a(488) which missed a trailing zero (see the table). - Philippe Guglielmetti, Jun 20 2017

A040017 Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime) of the form A019328(r)/gcd(A019328(r),r) in order (periods r are given in A051627).

Original entry on oeis.org

3, 11, 37, 101, 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991, 909090909090909090909090909091

Offset: 1

Jud McCranie

Prime p=3 is the only known example of a unique period prime such that A019328(r)/gcd(A019328(r),r) = p^k with k > 1 (cf. A323748). It is plausible to assume that no other such prime exists. Under this (unproved) assumption, the current sequence lists all unique period primes in order and represents a sorted version of A007615. - Max Alekseyev, Oct 14 2022

			The decimal expansion of 1/101 is 0.00990099..., having a period of 4 and it is the only prime with that period.

J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 324, Pour la Science Paris 2000.

Robert G. Wilson v, Table of n, a(n) for n = 1..47
Chris Caldwell, The Prime Glossary, Unique prime.
C. K. Caldwell, "Top Twenty" page, Unique.
Chris K. Caldwell and Harvey Dubner, Unique-Period Primes, J. Recreational Math., 29:1 (1998) 43-48.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Unique Prime.
Wikipedia, Unique prime.
Index entries for sequences related to decimal expansion of 1/n

Cf. A007615, A007498, A002371, A048595, A006883, A007732, A051626, A051627, A323748.

lst = {}; Do[c = Cyclotomic[n, 10]; q = c/GCD[c, n]; If[PrimeQ[q], AppendTo[lst, q]], {n, 62}]; Prepend[Sort[lst], 3] (* Arkadiusz Wesolowski, May 13 2012 *)

For n >= 2, a(n) = A019328(r) / gcd(A019328(r), r), where r = A051627(n). - Max Alekseyev, Oct 14 2022

Missing term a(45) inserted in b-file at the suggestion of Eric Chen by Max Alekseyev, Oct 13 2022

Edited by Max Alekseyev, Oct 14 2022

A007498 Unique period lengths of primes mentioned in A007615.

Original entry on oeis.org

1, 2, 3, 4, 9, 10, 12, 14, 19, 23, 24, 36, 38, 39, 48, 62, 93, 106, 120, 134, 150, 196, 294, 317, 320, 385, 586, 597, 654, 738, 945, 1031, 1172, 1282, 1404, 1426, 1452, 1521, 1752, 1812, 1836, 1844, 1862, 2134, 2232, 2264, 2667, 3750, 3903, 3927, 4274, 4354

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Let {Zs(m, 10, 1)} be the Zsigmondy numbers for a = 10, b = 1: Zs(m, 10, 1) is the greatest divisor of 10^m - 1^m that is coprime to 10^r - 1^r for all positive integers r < m. Then this sequence gives m such that Zs(m, 10, 1) is a prime power (e.g., Zs(1, 10, 1) = 9 = 3^2, Zs(2, 10, 1) = 11, Zs(3, 10, 1) = 37, Zs(4, 10, 1) = 101). It is very likely that Zs(m, 10, 1) is prime if m > 1 is in this sequence (note that the Mathematica and PARI programs below are based on this assumption). - Jianing Song, Aug 12 2020

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Samuel Yates, Period Lengths of Exactly One or Two Prime Numbers, J. Rec. Math., 18 (1985), 22-24.

Ray Chandler, Table of n, a(n) for n = 1..106
Chris K. Caldwell, Unique (period) primes and the factorization of cyclotomic polynomials minus one, Mathematica Japonica, 26 (1997), 189-195.
Chris K. Caldwell & H. Dubner, Unique-Period Primes, Table 2 in Journal of Recreational Mathematics 29(1) 46 1998.
Robert G. Wilson v, Notes, n.d.
Index entries for sequences related to decimal expansion of 1/n

Cf. A007615, A002371, A048595, A006883, A007732, A051626, A046108.

Cf. A161508 (unique period lengths in base 2).

lst={1}; Do[p=Cyclotomic[n, 10]/GCD[n, Cyclotomic[n, 10]]; If[PrimeQ[p], AppendTo[lst, n]], {n, 3000}]; lst (* T. D. Noe, Sep 08 2005 *)

isok(n) = if (n==1, 1, my(p = polcyclo(n, 10)); isprime(p/gcd(p, n))); \\ Michel Marcus, Jun 20 2018

More terms from T. D. Noe, Sep 08 2005

a(48)-a(52) from Ray Chandler, Jul 09 2008

A007615 Primes with unique period length (the periods are given in A007498).

Original entry on oeis.org

3, 11, 37, 101, 333667, 9091, 9901, 909091, 1111111111111111111, 11111111111111111111111, 99990001, 999999000001, 909090909090909091, 900900900900990990990991, 9999999900000001, 909090909090909090909090909091, 900900900900900900900900900900990990990990990990990990990991

Offset: 1

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Additional terms are Phi(n,10)/gcd(n,Phi(n,10)) for the n in A007498, where Phi(n,10) is the n-th cyclotomic polynomial evaluated at 10.

			3 is the only prime p such that decimal expansion of 1/p has (nontrivial) period exactly 1.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Samuel Yates, Period Lengths of Exactly One or Two Prime Numbers, J. Rec. Math., 18 (1985), 22-24.

Max Alekseyev, Table of n, a(n) for n = 1..98 (terms 1..25 from T. D. Noe; terms 26..31 from Ray Chandler)
C. K. Caldwell, The Prime Glossary, unique prime
Makoto Kamada, Factorizations of Phi_n(10)
Index entries for sequences related to decimal expansion of 1/n

Cf. A007498, A040017, A002371, A048595, A006883, A007732, A051626, A061075, A006530, A019328.

nmax = 50; periods = Reap[ Do[ p = Cyclotomic[n, 10] / GCD[n, Cyclotomic[n, 10]]; If[ PrimeQ[p], Sow[n]], {n, 1, nmax}]][[2, 1]]; Cyclotomic[#, 10] / GCD[#, Cyclotomic[#, 10]]& /@ periods // Prepend[#, 3]& (* Jean-François Alcover, Mar 28 2013 *)

a(n) = A061075(A007498(n)). - Max Alekseyev, Oct 16 2010

a(n) = A006530(A019328(A007498(n))). - Ray Chandler, May 10 2017

A048963 Table in which n-th row lists digits in periodic part of decimal expansion of reciprocal of n-th prime.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

The length of row n is A048595(n). - T. D. Noe, May 14 2008
The convention is that the earliest period is displayed. - T. D. Noe, May 14 2008
Conjecture: regarded as a decimal fraction, this number is normal in base 10. - Franklin T. Adams-Watters, Aug 20 2012

			1/2=.5 ->0; 1/3=.3333... -> 3; 1/5=.2 ->0; 1/7=.142857... -> 1 4 2 8 5 7; etc.
0; 3; 0; 1,4,2,8,5,7; 0,9; 7,6,9,2,3,0; 5,8,8,2,3,5,2,9,4,1,1,7,6,4,7,0; ...

Conway and Guy, The Book of Numbers, p. 160

Cf. A048962.

Clear[d]; d[{{2|5}, 0}] = {0}; d[{{{n__}}, 0}] := {n}; d[{{{n__, 0}}, k_?Negative}] := Join[Table[0, {-k}], Drop[{n}, k+1]]; A048963 = d /@ RealDigits[1/Prime[Range[10]]] (* Jean-François Alcover, Dec 10 2014 *)

A060370 Ratios (p-1)/d, where p is a prime and d is the number of digits of the periodic part of the decimal expansion of 1/p.

Original entry on oeis.org

1, 2, 4, 1, 5, 2, 1, 1, 1, 1, 2, 12, 8, 2, 1, 4, 1, 1, 2, 2, 9, 6, 2, 2, 1, 25, 3, 2, 1, 1, 3, 1, 17, 3, 1, 2, 2, 2, 1, 4, 1, 1, 2, 1, 2, 2, 7, 1, 2, 1, 1, 34, 8, 5, 1, 1, 1, 54, 4, 10, 2, 2, 2, 2, 1, 4, 3, 1, 2, 3, 11, 2, 1, 2, 1, 1, 1, 4, 2, 2, 1, 3, 2, 1, 2

Offset: 1

Klaus Brockhaus, Apr 01 2001

The sequence of 2nd, 4th and following terms coincides with A006556, which gives the "number of different cycles of digits in the decimal expansions of 1/p, 2/p, ..., (p-1)/p where p = n-th prime different from 2 or 5".

			a(13) = 40/5 = 8, since 41 is the 13th prime and the periodic part of 1/41 = 0.02439024390... consists of 5 digits.

Carmine Suriano and T. D. Noe, Table of n, a(n) for n = 1..10000 (first 101 terms from Carmine Suriano)

Cf. A000040, A060283, A048595, A006556.

Join[{1, 2, 4}, Table[p = Prime[n]; (p - 1)/Length[RealDigits[1/p, 10][[1, 1]]], {n, 4, 100}]] (* T. D. Noe, Oct 04 2012 *)

from sympy import prime, n_order
def A060370(n): return 1 if n == 1 or n == 3 else n_order(10, prime(n))
print([(prime(n)-1)//A060370(n) for n in range(1,86)]) # Karl-Heinz Hofmann, Mar 16 2022

a(n) = (b(n)-1)/c(n), where b(n) and c(n) are the n-th terms of A000040 and A048595 respectively.

A186635 Primes p such that the decimal expansion of 1/p has a periodic part of odd length.

Original entry on oeis.org

2, 3, 5, 31, 37, 41, 43, 53, 67, 71, 79, 83, 107, 151, 163, 173, 191, 199, 227, 239, 271, 277, 283, 307, 311, 317, 347, 359, 397, 431, 439, 443, 467, 479, 523, 547, 563, 587, 599, 613, 631, 643, 683, 719, 733, 751, 757, 773, 787, 797, 827, 839, 853, 883, 907, 911, 919, 947, 991, 1013, 1031, 1039, 1093, 1123, 1151, 1163, 1187

Offset: 1

Jani Melik, Feb 24 2011

Interestingly, the initial terms of A040119 (Primes p such that x^4 = 10 has a solution mod p) are identical to the initial terms of this sequence except for 241 which is a term of A040119 but not of A186635. [John W. Layman, Feb 25 2011]

There are many numbers in A040119 that are not here: 241, 641, 769, 809, 1009, 1409, 1601, 1721.... - T. D. Noe, Feb 25 2011

Cf. A002371, A048595, A028416 (complement in the primes), A040119.

Ax := proc(n) local st:
st := ithprime(n):
if (modp(numtheory[order](10,st),2) <> 0) then
   RETURN(st)
fi: end:  seq(Ax(n), n=1..200);

Union[{2, 5}, Select[Prime[Range[200]], OddQ[Length[RealDigits[1/#][[1, 1]]]] &]]

select( {is_A186635(n)=isprime(n) && (n<7 || znorder(Mod(10, n))%2)}, [0..1234]) \\ M. F. Hasler, Nov 19 2024

from sympy import isprime, n_order
is_A186635 = lambda n: isprime(n) and (n<7 or n_order(10, n)%2)
[n for n in range(1234) if is_A186635(n)] # M. F. Hasler, Nov 19 2024

cp's OEIS Frontend

A002371 Period of decimal expansion of 1/(n-th prime) (0 by convention for the primes 2 and 5).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions

A051626 Period of decimal representation of 1/n, or 0 if 1/n terminates.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Python

Formula

Extensions

A006556 Number of different cycles of digits in the decimal expansions of 1/p, 2/p, ..., (p-1)/p where p = n-th prime different from 2 or 5.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A036275 The periodic part of the decimal expansion of 1/n. Any initial 0's are to be placed at end of cycle.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Extensions

A040017 Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime) of the form A019328(r)/gcd(A019328(r),r) in order (periods r are given in A051627).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A007498 Unique period lengths of primes mentioned in A007615.

Views

Author

Keywords

Comments

References

Links

Crossrefs