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.

Previous Showing 11-17 of 17 results.

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

A098680 Primes p whose period of reciprocal equals (p-1)/13.

Original entry on oeis.org

2393, 15497, 18149, 18617, 20021, 25819, 26183, 26339, 29303, 39937, 42953, 48491, 52313, 53327, 57331, 58189, 59021, 65183, 81953, 82499, 87491, 91703, 98047, 102233, 104287, 109097, 111229, 119419, 129793, 131171, 143287, 143833, 162007
Offset: 1

Views

Author

Robert G. Wilson v, Sep 15 2004

Keywords

Links

Crossrefs

Cf. A097443, A055628, A056157, A056210, A056211, A056212, A056213, A056214, A056215, A056216, A056217, 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, 15000]], f[ # ] == 13 &]

A135073 Primes for which the period of the reciprocal equals (p-1)/14.

Original entry on oeis.org

449, 1289, 3557, 4397, 4999, 5209, 6203, 6637, 7043, 8387, 10613, 11369, 13147, 13399, 14323, 16871, 18481, 19391, 20147, 20707, 26489, 28813, 29387, 29947, 30241, 32831, 32999, 33587, 36107, 37591, 38053, 39719, 40559, 41231, 41609
Offset: 1

Views

Author

Julien Peter Benney (jpbenney(AT)gmail.com), Feb 12 2008

Keywords

Comments

Also cyclic numbers of the fourteenth degree (or fourteenth order): the reciprocals of these numbers belong to one of fourteen different cycles. Each cycle has the (number minus 1)/14 digits.

Examples

			1289 has period of reciprocal 92, or (1289/1)/14.

Links

Crossrefs

Cf. A097443, A055628, A056157, A056210, A056211, A056212, A056213, A056214, A056215, A056216, A056217, A098680.

Programs

  • Maple
    A007732 := proc(n) local nred25 ; nred25 := n ; while nred25 mod 2 = 0 and nred25 > 1 do nred25 := nred25/2 ; od; while nred25 mod 5 = 0 and nred25 > 1 do nred25 := nred25/5 ; od; if nred25 = 1 then 1; else numtheory[order](10,nred25) ; fi ; end: for n from 1 to 22000 do p := ithprime(n) ; if 14*A007732(p) = p-1 then printf("%d,",p) ; fi ; od: # R. J. Mathar, Feb 21 2008
  • Mathematica
    Select[Prime[Range[4500]],Length[RealDigits[1/#][[1,1]]]==(#-1)/14&] (* Harvey P. Dale, Jun 22 2013 *)

Extensions

Corrected and extended by R. J. Mathar, Feb 21 2008

A007348 Primes for which -10 is a primitive root.

Original entry on oeis.org

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

Views

Author

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

Keywords

Comments

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

References

  • 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).

Links

Crossrefs

Cf. A038880.
Cf. A006883, A097443.

Programs

  • Mathematica
    pr=-10; Select[Prime[Range[200 ] ], MultiplicativeOrder[pr, # ] == #-1 & ]
  • PARI
    is(n)=gcd(n,10)==1 && znorder(Mod(-10,n))==n-1 \\ Charles R Greathouse IV, Nov 25 2014

Extensions

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

A381590 Primes with primitive root -100.

Original entry on oeis.org

3, 7, 19, 23, 31, 43, 47, 59, 67, 71, 83, 107, 131, 151, 163, 167, 179, 191, 199, 223, 227, 263, 283, 307, 311, 347, 359, 367, 379, 383, 419, 431, 439, 443, 467, 479, 487, 491, 499, 503, 523, 563, 571, 587, 599, 619, 631, 647, 659, 683, 719, 727, 743, 787, 811
Offset: 1

Views

Author

Davide Rotondo, Feb 28 2025

Keywords

Comments

Union of long period primes (A006883) of the form 4k-1 and half period primes (A097443) of the form 4k-1.
Complement of A007349 in the union of A007348 and A001913. - Davide Rotondo, May 23 2025

Crossrefs

Cf. A006883, A097443, A007349, A007348, A001913.

Programs

  • Mathematica
    Select[Prime[Range[150]], MultiplicativeOrder[-100, #] == # - 1 &]  (* Amiram Eldar, Mar 02 2025 *)
  • PARI
    is(n)=gcd(n,10)==1 && znorder(Mod(-100, n))==n-1 \\ Charles R Greathouse IV, Mar 01 2025
  • PARI
    list(lim)=my(v=List([3])); forprime(p=7,lim, if(znorder(Mod(-100, p))==p-1, listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Mar 01 2025
Previous Showing 11-17 of 17 results.

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

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