cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 12 results. Next

A098672 Duplicate of A056210.

Original entry on oeis.org

11, 251, 1061, 1451, 1901, 1931, 2381, 3181, 3491, 3851, 4621, 4861, 5261, 6101
Offset: 1

Views

Author

Keywords

A097443 Half-period primes, i.e., primes p for which the decimal expansion of 1/p has period (p-1)/2.

Original entry on oeis.org

3, 13, 31, 43, 67, 71, 83, 89, 107, 151, 157, 163, 191, 197, 199, 227, 283, 293, 307, 311, 347, 359, 373, 401, 409, 431, 439, 443, 467, 479, 523, 557, 563, 569, 587, 599, 601, 631, 653, 677, 683, 719, 761, 787, 827, 839, 877, 881, 883, 911, 919, 929, 947, 991
Offset: 1

Views

Author

Julien Peter Benney (jpbenney(AT)ftml.net), Aug 23 2004

Keywords

Comments

Primes p for which 10 has multiplicative order (p-1)/2. - Robert Israel, Jul 15 2016

Examples

			13 is a half-period prime because 1/13 = 0.076923076923076923076923..., which has period 6, or (13-1)/2.

Links

Crossrefs

Almost the same as A001914.
Cf. A001913, A055628, A056157, A056210-A056217, A098680
Cf. also A000040, A007732, A006883, A097443, A097955.

Programs

  • Maple
    select(t -> isprime(t) and numtheory:-order(10, t) = (t-1)/2,
[seq(t,t = 3..1000,2)]); # Robert Israel, Jul 15 2016
  • Mathematica
    f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 200]], f[ # ] == 2 &] (* Robert G. Wilson v, Sep 14 2004 *)
  • PARI
    is(n)= gcd(10,n)==1 && isprime(n) && znorder(Mod(10,n))==(n-1)/2 \\ Dana Jacobsen, Jul 19 2016
  • Perl
    use ntheory ":all"; forprimes { say if znorder(10,$) == ($-1)/2; } 1,1000; # Dana Jacobsen, Jul 19 2016

Extensions

Edited (including prepending 3), at the suggestion of Georg Fischer, by N. J. A. Sloane, Oct 19 2018

A056157 Primes p whose period of reciprocal equals (p-1)/4.

Original entry on oeis.org

53, 173, 277, 317, 397, 769, 773, 797, 809, 853, 1009, 1013, 1093, 1493, 1613, 1637, 1693, 1721, 2129, 2213, 2333, 2477, 2521, 2557, 2729, 2797, 2837, 3329, 3373, 3517, 3637, 3733, 3797, 3853, 3877, 4133, 4241, 4253, 4373, 4493, 4729, 4733, 4877, 5081
Offset: 1

Views

Author

Don Willard (dwillard(AT)prairie.cc.il.us), Jun 05 2000

Keywords

Comments

Cyclic numbers of the fourth degree (or fourth order): the reciprocals of these numbers belong to one of four different cycles. Each cycle has the (number minus 1)/4 digits.

Links

Crossrefs

Cf. A001913, A097443, A055628, A056210, A056211, A056212, A056213, A056214, A056215, A056216, A056217, A098680.

Programs

  • Mathematica
    f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 700]], f[ # ] == 4 &] (* Robert G. Wilson v, Aug 02 2000 *)
LP[ n_Integer ] := (ds = Divisors[ n - 1 ]; Take[ ds, Position[ PowerMod[ 10, ds, n ], 1 ][ [ 1, 1 ] ] ][ [ -1 ] ]); CL[ n_Integer ] := (n - 1)/LP[ n ]; Select[ Range[ 7, 7500 ], PrimeQ[ # ] && CL[ # ] == 4 & ] (* Robert G. Wilson v, Aug 02 2000 *)

Extensions

More terms from Robert G. Wilson v, Aug 02 2000

A055628 Primes p whose period of the reciprocal 1/p is (p-1)/3.

Original entry on oeis.org

103, 127, 139, 331, 349, 421, 457, 463, 607, 661, 673, 691, 739, 829, 967, 1657, 1669, 1699, 1753, 1993, 2011, 2131, 2287, 2647, 2659, 2749, 2953, 3217, 3229, 3583, 3691, 3697, 3739, 3793, 3823, 3931, 4273, 4297, 4513, 4549, 4657, 4903, 4909, 4993, 5011
Offset: 1

Views

Author

Don Willard (dwillard(AT)prairie.cc.il.us), Jun 05 2000

Keywords

Comments

Cyclic numbers of the third degree (or third order): the reciprocals of these numbers belong to one of three different cycles. Each cycle has (number-1)/3 digits.
All primes p except 2 or 5 have a reciprocal with period which divides p-1.

Examples

			127 has period 42 and (127-1)/3 = 126/3 = 42.

References

  • Stephen P. Richards, A Number For Your Thoughts, 1982, 1984, Box 501, New Providence, NJ, 07974, ISBN 0-9608224-0-2.

Links

Crossrefs

Cf. A054471, A001914, A001913, A097443, A056157, A056210, A056211, A056212, A056213, A056214, A056215, A056216, A056217, A098680.

Programs

  • Mathematica
    LP[ n_Integer ] := (ds = Divisors[ n - 1 ]; Take[ ds, Position[ PowerMod[ 10, ds, n ], 1 ][ [ 1, 1 ] ] ][ [ -1 ] ]); CL[ n_Integer ] := (n - 1)/LP[ n ]; Select[ Range[ 7, 7500 ], PrimeQ[ # ] && CL[ # ] == 3 & ]
f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 700]], f[ # ] == 3 &] (* Robert G. Wilson v, Sep 14 2004 *)

Extensions

More terms from Robert G. Wilson v, Aug 02 2000
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007

A054471 Smallest prime p having n different cycles in decimal expansions of k/p, k=1..p-1.

Original entry on oeis.org

7, 3, 103, 53, 11, 79, 211, 41, 73, 281, 353, 37, 2393, 449, 3061, 1889, 137, 2467, 16189, 641, 3109, 4973, 11087, 1321, 101, 7151, 7669, 757, 38629, 1231, 49663, 12289, 859, 239, 27581, 9613, 18131, 13757, 33931, 9161, 118901, 6763, 18233
Offset: 1

Views

Author

Robert G. Wilson v, 1994; Antreas P. Hatzipolakis (xpolakis(AT)otenet.gr), May 22 2000

Keywords

Comments

First cyclic number of n-th degree (or n-th order): the reciprocals of these numbers belong to one of n different cycles. Each cycle has (a(n) - 1)/n digits.
From Robert G. Wilson v, Aug 21 2014: (Start)
recursive by indices:
1, 7, 211, 79337, 634776923741, ...
2, 3, 103, 2368589, 785245568161181, ...
4, 53, 135257, 2332901103899, ...
5, 11, 353, 3795457, 693814982285339, ...
6, 79, 26861, 23947548497, ...
8, 41, 118901, 1015118238709, ...
9, 73, 142789, 267291583927, ...
10, 281, 3097183, 66880786504811, ...
12, 37, 18131, 105385168331, ...
13, 2393, 11160953, 7140939250711817, ...
14, 4999, 2148340247, > 10^19,
... .
(End)

References

  • John H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, p. 162.
  • M. Gardner, Mathematical Circus, Cambridge University Press (1996).

Links

Crossrefs

First time n appears in A006556.
Cf. A006883, A097443, A055628, A056157, A056210, A056211, A056212, A056213, A056214, A056215, A056216, A056217, A098680, which are sequences of primes p where the period of the reciprocal is (p-1)/n for n=1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13.
Cf. A101208, A101209 (similar sequences for base 2 and base 3).

Programs

  • Mathematica
    a[n_Integer] := Block[{m = If[ OddQ@ n, 2n, n]}, p = m +1; While[ !PrimeQ@ p || p != 1 + n*MultiplicativeOrder[10, p], p = p += m]; p]; a[1] = 7; a[4] = 53; Array[f, 50] (* Robert G. Wilson v, Apr 19 2005; revised Aug 20 2014 and Feb 14 2025 *)

Extensions

More terms from David W. Wilson, May 22 2000

A056211 Primes p whose period of reciprocal equals (p-1)/6.

Original entry on oeis.org

79, 547, 643, 751, 907, 997, 1201, 1213, 1237, 1249, 1483, 1489, 1627, 1723, 1747, 1831, 1879, 1987, 2053, 2551, 2683, 3049, 3253, 3319, 3613, 3919, 4159, 4507, 4519, 4801, 4813, 4831, 4969, 5119, 5443, 5557, 5791, 6079, 6151, 6271, 6373, 6427, 6529
Offset: 1

Views

Author

Robert G. Wilson v, Aug 02 2000

Keywords

Comments

Cyclic numbers of the sixth degree (or sixth order): the reciprocals of these numbers belong to one of six different cycles. Each cycle has the (number minus 1)/6 digits.

Links

Crossrefs

Cf. A001913, A097443, A055628, A056157, A056210, A056212, A056213, A056214, A056215, A056216, A056217, A098680.

Programs

  • Mathematica
    f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 850]], f[ # ] == 6 &]

Extensions

Edited by N. J. A. Sloane, Apr 30 2007

A056212 Primes p whose period of reciprocal equals (p-1)/7.

Original entry on oeis.org

211, 617, 1499, 2087, 2857, 6007, 6469, 7127, 7211, 7589, 9661, 10193, 13259, 13553, 14771, 18047, 18257, 19937, 20903, 21379, 23549, 26153, 27259, 27539, 32299, 33181, 33461, 34847, 35491, 35897, 41651, 42407, 42491, 43051, 43793
Offset: 1

Views

Author

Robert G. Wilson v, Aug 02 2000

Keywords

Comments

Cyclic numbers of the seventh degree (or seventh order): the reciprocals of these numbers belong to one of seven different cycles. Each cycle has the (number minus 1)/7 digits.

Links

Crossrefs

Cf. A097443, A055628, A056157, A056210, A056211, A056213, A056214, A056215, A056216, A056217, A098680, A135073.

Programs

  • Mathematica
    f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 4700]], f[ # ] == 7 &]

Extensions

Edited by N. J. A. Sloane, Apr 30 2007

A056214 Primes p whose period of reciprocal equals (p-1)/9.

Original entry on oeis.org

73, 1423, 1459, 2377, 2503, 3457, 7741, 9433, 10891, 10909, 16057, 17299, 17623, 20269, 21313, 22699, 24103, 26263, 28621, 28927, 29629, 30817, 32257, 34273, 34327, 35461, 35731, 36343, 36793, 37549, 37567, 37657, 38737, 39367, 39979
Offset: 1

Views

Author

Robert G. Wilson v, Aug 02 2000

Keywords

Comments

Cyclic numbers of the ninth degree (or ninth order): the reciprocals of these numbers belong to one of nine different cycles. Each cycle has the (number minus 1)/9 digits.

Links

Crossrefs

Cf. A097443, A055628, A056157, A056210, A056211, A056212, A056213, A056215, A056216, A056217, A098680, A135073.

Programs

  • Mathematica
    f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 5000]], f[ # ] == 9 &]

Extensions

Edited by N. J. A. Sloane, Apr 30 2007

A056215 Primes p whose reciprocal has period (p-1)/10.

Original entry on oeis.org

281, 521, 1031, 1951, 2281, 2311, 2591, 3671, 5471, 5711, 6791, 7481, 8111, 8681, 8761, 9281, 9551, 10601, 11321, 12401, 13151, 13591, 14831, 14951, 15671, 16111, 16361, 18671, 21191, 21521, 21881, 24281, 24551, 25391, 25801, 25841, 26161
Offset: 1

Views

Author

Robert G. Wilson v, Sep 15 2004

Keywords

Comments

Cyclic numbers of the tenth degree (or tenth order): the reciprocals of these numbers belong to one of ten different cycles. Each cycle has (p-1)/10 digits.

Links

Crossrefs

Cf. A097443, A055628, A056157, A056210, A056211, A056212, A056213, A056214, A056216, A056217, A098680, A135073.

Programs

  • Mathematica
    f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 3000]], f[ # ] == 10 &]

Extensions

Entry revised by N. J. A. Sloane, Apr 30 2007

A056216 Primes p whose period of reciprocal equals (p-1)/11.

Original entry on oeis.org

353, 3499, 10429, 13619, 15269, 20219, 20593, 23057, 23189, 24091, 25741, 30713, 35509, 38567, 45233, 49171, 57179, 57223, 60149, 63691, 63977, 67783, 77023, 85229, 88463, 90619, 91367, 93941, 96779, 108967, 109913, 110221, 112069
Offset: 1

Views

Author

Robert G. Wilson v, Aug 02 2000

Keywords

Comments

Cyclic numbers of the eleventh degree (or eleventh order): the reciprocals of these numbers belong to one of eleven different cycles. Each cycle has the (number minus 1)/11 digits.

Links

Crossrefs

Cf. A097443, A055628, A056157, A056210, A056211, A056212, A056213, A056214, A056215, A056217, A098680, A135073.

Programs

  • Mathematica
    f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 11000]], f[ # ] == 11 &]

Extensions

Edited by N. J. A. Sloane, Apr 30 2007
Showing 1-10 of 12 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.