A006883 -id:A006883 - cp's OEIS frontend

A007348 Primes for which -10 is a primitive root.

3, 17, 29, 31, 43, 61, 67, 71, 83, 97, 107, 109, 113, 149, 151, 163, 181, 191, 193, 199, 227, 229, 233, 257, 269, 283, 307, 311, 313, 337, 347, 359, 389, 431, 433, 439, 443, 461, 467, 479, 509, 523, 541, 563, 577, 587, 593, 599, 631, 683, 701, 709, 719, 787, 821, 827, 839

Offset: 1

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

nonn

Union of long period primes (A006883) of the form 4k+1 and half period primes (A097443) of the form 4k+3. - Davide Rotondo, Aug 25 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 24.8, p. 864.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. J. Mathar, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993
Index entries for primes by primitive root

Cf. A038880.

Cf. A006883, A097443.

pr=-10; Select[Prime[Range[200 ] ], MultiplicativeOrder[pr, # ] == #-1 & ]

is(n)=gcd(n,10)==1 && znorder(Mod(-10,n))==n-1 \\ Charles R Greathouse IV, Nov 25 2014

More terms from N. J. A. Sloane, Apr 24 2005
Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar
A&S reference and Mathematica program corrected by T. D. Noe, Nov 04 2009

A006559 Short period primes: the decimal expansion of 1/p has period less than p-1, but greater than zero.

Original entry on oeis.org

3, 11, 13, 31, 37, 41, 43, 53, 67, 71, 73, 79, 83, 89, 101, 103, 107, 127, 137, 139, 151, 157, 163, 173, 191, 197, 199, 211, 227, 239, 241, 251, 271, 277, 281, 283, 293, 307, 311, 317, 331, 347, 349, 353, 359, 373, 397, 401, 409, 421, 431, 439, 443, 449, 457

Offset: 1

N. J. A. Sloane

Primes 2 and 5 are excluded because 1/2 and 1/5 have no period. Also primes p whose multiplicative order mod p is less than p-1.

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 = 1..1000
Victor Meally, Letter to N. J. A. Sloane, no date.
Index entries for sequences related to decimal expansion of 1/n

Cf. A006883.

Select[Prime[Range[100]], MultiplicativeOrder[10, #] < # - 1 &]

a(n)=gcd(n,10)==1 && isprime(n) && znorder(Mod(10,n))Charles R Greathouse IV, Mar 15 2014

from itertools import islice
from sympy import nextprime, n_order
def A006559_gen(startvalue=1): # generator of terms >= startvalue
    p = max(startvalue-1,1)
    while (p:=nextprime(p)):
        if p!=2 and p!=5 and n_order(10,p)A006559_list = list(islice(A006559_gen(),20)) # Chai Wah Wu, Mar 03 2025

More terms from James Sellers, Aug 21 2000

A180340 Numbers with x digits such that the first x multiples are cyclic permutations of the number, leading 0's omitted (or cyclic numbers).

Original entry on oeis.org

142857, 588235294117647, 52631578947368421, 434782608695652173913, 344827586206896551724137931, 212765957446808510638297872340425531914893617, 169491525423728813559322033898305084745762711864406779661

Offset: 1

Ralph Kerchner (daxkerchner(AT)hotmail.com), Aug 28 2010

Periodic part of decimal expansion of 1/A001913(n). The number of digits in each term (including leading zeros), plus one, makes the sequence A001913.

			142857 is in the sequence because it has 6 digits and the first 6 multiples of 142857 are 142857, 285714, 428571, 571428, 714285, and 857142, all cyclic permutations of the number. Also the first term of A001913 is 7, and 1/7 = 0.142857142857... .
588235294117647 is the next number because 0588235294117647 has 16 digits and the first 16 multiples are cyclic permutations of the number; the second term of A001913 is 17, and 1/17 = 0.05882352941176470588235294117647... .

Ray Chandler, Table of n, a(n) for n = 1..60
Edwin E. Freed, Binary Magic Numbers, Dr. Dobb's Journal, Vol. 78 (April 1983), pp. 24-37.
OEIS Wiki, Cyclic numbers
Eric Weisstein's World of Mathematics, Cyclic number
Wikipedia, Cyclic number

A006883 starting from the second term of A006883, omitting ending 0's.

The n-th terms of A060284 where n is a member of A001913.

Map[(10^(# - 1) - 1)/# &, Select[Prime@ Range@ 17, MultiplicativeOrder[10, #] == # - 1 &]] (* Michael De Vlieger, Apr 03 2017 *)

a(n) = (10^(A001913(n)-1) - 1) / A001913(n).

A221981 Primes q = 4*p+1, where p == 2 (mod 5) is also prime.

Original entry on oeis.org

29, 149, 269, 389, 509, 1109, 1229, 1949, 2309, 2909, 3989, 4349, 5189, 5309, 6269, 6389, 7109, 7949, 8069, 9749, 10589, 10709, 11069, 11549, 12149, 12269, 13229, 13829, 14549, 15629, 16229, 17189, 17789, 18269, 19949, 20789, 22109, 22229, 24029, 24989, 25349, 25469, 25589, 26189, 26309, 28109, 28229, 28949, 29669, 30029, 30869, 31469, 32069, 33149, 34589, 34949, 36269, 36629, 36749, 37589

Offset: 1

Jonathan Sondow, Feb 02 2013

nonn

Moree (2012) says that Chebyshev observed that if q = 4p + 1 is prime, with prime p == 2 (mod 5), then 10 is a primitive root modulo q.
If the sequence is infinite, then Artin's conjecture ("every nonsquare integer n != -1 is a primitive root of infinitely many primes q") is true for n = 10.
The corresponding primes p are A221982.
The sequence is infinite under Dickson's conjecture, thus Dickson's conjecture implies Artin's conjecture for n = 10. - Charles R Greathouse IV, Apr 18 2013
Two conjectures: (a) These primes have primitive root 40; (b) if a(n)*8 + 1 is prime then it has primitive root 10. - Davide Rotondo, Dec 31 2024

			29 is a member because 29 = 4*7 + 1 and 7 == 2 (mod 5) are prime.

P. L. Chebyshev, Theory of congruences, Elements of number theory, Chelsea, 1972, p. 306.
Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section F9, pp. 377-380.

Amiram Eldar, Table of n, a(n) for n = 1..10000
P. L. Chebyshev, Theorie der Congruenzen, Mayer & Mueller, 1889, pp. 306-313.
Pieter Moree, Artin's primitive root conjecture - a survey, arXiv:math/0412262 [math.NT], 2004, revised 2012, p. 1.
Index entries for sequences related to Artin's conjecture
Index entries for primes by primitive root

Cf. A001913, A005596, A006883, A045380, A106849, A221982, A222008.

A221981:=n->`if`(isprime(4*n+1) and isprime(n) and n mod 5 = 2, 4*n+1, NULL): seq(A221981(n), n=1..10^4); # Wesley Ivan Hurt, Dec 11 2015

Select[ Prime[ Range[4000]], Mod[(# - 1)/4, 5] == 2 && PrimeQ[(# - 1)/4] &]

is(n)=n%20==9 && isprime(n) && isprime(n\4) \\ Charles R Greathouse IV, Apr 18 2013

a(n) = 4*A221982(n) + 1.

a(n) >> n log^2 n. - Charles R Greathouse IV, Dec 30 2024

A000353 Primes p == 7, 19, 23 (mod 40) such that (p-1)/2 is also prime.

Original entry on oeis.org

7, 23, 47, 59, 167, 179, 263, 383, 503, 863, 887, 983, 1019, 1367, 1487, 1619, 1823, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2903, 3023, 3167, 3623, 3779, 3863, 4007, 4127, 4139, 4259, 4703, 5087, 5099, 5807, 5927, 5939, 6047, 6659, 6779, 6899, 6983, 7247

Offset: 1

Robert A. J. Matthews

nonn

The decimal expansion of 1/a(n) will produce a stream of a(n)-1 pseudo-random digits. - Reinhard Zumkeller, Feb 10 2009

The condition in the name is sufficient for primes p such that the decimal expansion of 1/p recurs after p-1 digits, which is the maximum-possible cycle length. - Robert A. J. Matthews, Oct 31 2023

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Robert A. J. Matthews, Maximally periodic reciprocals, Bull. Institute of Mathematics and Its Applications, vol. 28, p. 147-148, 1992.
Wikipedia, Sophie Germain prime
Index entries for sequences related to decimal expansion of 1/n.

Subset of A005385.

Subsequence of A001913, A006883.

q:= p-> irem(p, 40) in {7, 19, 23} and andmap(isprime, [p, (p-1)/2]):
select(q, [$1..10000])[];  # Alois P. Heinz, Oct 31 2023

Select[Prime[Range[1000]], MatchQ[Mod[#, 40], 7|19|23] && PrimeQ[(#-1)/2]&] (* Jean-François Alcover, Feb 07 2016 *)

is(n)=my(k=n%40); (k==7||k==19||k==23) && isprime(n\2) && isprime(n) \\ Charles R Greathouse IV, Nov 20 2014

a(n) = 2*A000355(n)+1. - Reinhard Zumkeller, Feb 10 2009

More terms from Reinhard Zumkeller, Feb 10 2009

A007349 Primes with both 10 and -10 as primitive root.

Original entry on oeis.org

17, 29, 61, 97, 109, 113, 149, 181, 193, 229, 233, 257, 269, 313, 337, 389, 433, 461, 509, 541, 577, 593, 701, 709, 821, 857, 937, 941, 953, 977, 1021, 1033, 1069, 1097, 1109, 1153, 1181, 1193, 1217, 1229, 1297, 1301, 1381, 1429, 1433, 1549, 1553, 1621, 1697, 1709, 1741, 1777, 1789, 1861, 1873, 1913, 1949

Offset: 1

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

nonn

Intersection of A006883 and A002144. - Davide Rotondo, May 21 2025

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 864.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Robert G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993
Index entries for primes by primitive root

Cf. A002144, A006883.

pr=10; Select[Prime[Range[200]], MultiplicativeOrder[pr, # ] == MultiplicativeOrder[-pr, # ] == #-1 &]
Select[Prime[Range[5,200]],PrimitiveRoot[#,10]==10&&PrimitiveRoot[#,#-10] == #-10&] (* Harvey P. Dale, Oct 10 2019 *)

forprime(p=11,2000,if(znorder(Mod(10,p))==p-1&&znorder(Mod(-10,p))==p-1,print1(p,", "))); \\ Joerg Arndt, May 21 2025

A221982 Primes p == 2 (mod 5) for which 4*p+1 is also prime.

Original entry on oeis.org

7, 37, 67, 97, 127, 277, 307, 487, 577, 727, 997, 1087, 1297, 1327, 1567, 1597, 1777, 1987, 2017, 2437, 2647, 2677, 2767, 2887, 3037, 3067, 3307, 3457, 3637, 3907, 4057, 4297, 4447, 4567, 4987, 5197, 5527, 5557, 6007, 6247, 6337, 6367, 6397, 6547, 6577, 7027, 7057, 7237, 7417, 7507, 7717, 7867

Offset: 1

Jonathan Sondow, Feb 02 2013

nonn

The corresponding primes 4*p+1 are Chebyshev's subsequence A221981 of the primes with primitive root 10.

			7 is a member because 7 == 2 (mod 5) and 29 = 4*7 + 1 are both prime.

P. L. Chebyshev, Theory of congruences. Elements of number theory, Chelsea, 1972, p. 306.
R. K. Guy, Unsolved Problems in Number Theory, F9.

Paolo P. Lava, Table of n, a(n) for n = 1..10000
P. Moree, Artin's primitive root conjecture - a survey, arXiv 2004, revised 2012, p. 1.
Index entries for primes by primitive root

Cf. A001913, A006883, A045380, A106849, A221981, A222008.

A221982:=proc(q)
local n;
for n from 1 to q do
if isprime(n) and isprime(4*n+1) and (n mod 5)=2 then print(n) fi; od; end:
A221982 (10000); # Paolo P. Lava, Feb 12 2013

Select[ Prime[ Range[1000]], Mod[#, 5] == 2 && PrimeQ[4 # + 1] &]

a(n) = (A221981(n) - 1)/4.

A056055 Integers k > 1 such that the decimal expansion of 1/k contains k as a string. (If the decimal expansion terminates, trailing zeros do not count.)

Original entry on oeis.org

3, 6, 7, 14, 17, 28, 58, 59, 83, 86, 87, 89, 97, 118, 167, 197, 228, 281, 313, 316, 339, 367, 379, 383, 456, 458, 469, 529, 541, 543, 569, 577, 587, 593, 607, 618, 626, 629, 647, 669, 673, 677, 678, 683, 687, 701, 709, 719, 722, 727, 729, 767, 771, 772, 778

Offset: 1

Ulrich Schimke (ulrschimke(AT)aol.com)

The sequence is probably infinite, since long-period primes (cf. A006883) especially with high first digit are likely candidates, but is there a proof? Does any k with finite expansion of 1/k (i.e., k = 2^j * 5^m) occur?

			118 is a term since 1/118 = 0.00847457627118... contains "118".
100 is not a term because 1/100 = 0.01 does not contain "100" (0.0100 does not count).

Nicolas Bělohoubek, Table of n, a(n) for n = 1..900
Index entries for sequences related to decimal expansion of 1/n

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

A067615 Terms in decimal expansion of 1/(17*2^n) before 5882352941176470 (the period with no leading zeros of 1/17) appears.

Original entry on oeis.org

2941176470, 1470, 7352941176470, 3676470, 18382352941176470, 9191176470, 4595, 22977941176470, 11488970, 57444852941176470, 28722426470, 143612132352941176470, 71806066176470, 359030330882352941176470

Offset: 1

Benoit Cloitre, Feb 01 2002; revised Dec 11, 2004

			Example: 1/(17*2^3) = 0.007352941176470(5882352941176470...) where 5882352941176470 is the period with no leading zero of 1/17 hence a(3)= 7352941176470

Cf. A067556, A006883.

cp's OEIS Frontend

A007348 Primes for which -10 is a primitive root.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A006559 Short period primes: the decimal expansion of 1/p has period less than p-1, but greater than zero.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

Python

Extensions

A180340 Numbers with x digits such that the first x multiples are cyclic permutations of the number, leading 0's omitted (or cyclic numbers).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A221981 Primes q = 4*p+1, where p == 2 (mod 5) is also prime.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A000353 Primes p == 7, 19, 23 (mod 40) such that (p-1)/2 is also prime.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A007349 Primes with both 10 and -10 as primitive root.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

PARI

A221982 Primes p == 2 (mod 5) for which 4*p+1 is also prime.

Views

Author

Keywords

Comments