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 1-6 of 6 results.

A127438 Minimal nonnegative solution to 2^x == 3 (mod p) where p goes over primes for which such a solution exists (A001915).

Original entry on oeis.org

0, 3, 8, 4, 13, 8, 5, 26, 19, 17, 50, 6, 39, 16, 72, 19, 69, 70, 72, 41, 87, 101, 48, 27, 108, 56, 89, 42, 181, 43, 46, 48, 83, 109, 157, 93, 92, 56, 249, 152, 26, 69, 238, 137, 184, 271, 11, 100, 404, 13, 318, 111, 450, 25, 320, 151, 130, 9, 297, 104, 429, 435, 530, 105, 478, 175, 114, 75, 175, 80
Offset: 1

Views

Author

Max Alekseyev, Jan 14 2007

Keywords

Crossrefs

Cf. A001915.

Programs

  • Magma
    lst:=[0]; for p in [5..647 by 2] do if IsPrime(p) then e:=Ceiling(Log(2, p+1)); for x in [e..p-2] do if 2^x mod p eq 3 then Append(~lst, x); break; end if; end for; end if; end for; lst; // Arkadiusz Wesolowski, Jan 12 2021

Formula

2^a(n) == 3 (mod A001915(n)), where a(n) >= 0 and minimum possible.

Extensions

Corrected by Max Alekseyev, Jun 08 2011
Corrected by Arkadiusz Wesolowski, Jan 12 2021

A127437 Duplicate of A001915.

Original entry on oeis.org

2, 5, 11, 13, 19, 23, 29, 37, 47, 53, 59, 61, 67, 71, 83, 97, 101, 107, 131, 139, 149, 163, 167, 173, 179, 181, 191, 193, 197, 211, 227, 239, 263, 269, 293, 307, 311, 313, 317, 347, 349, 359, 373, 379, 383, 389, 409, 419, 421, 431, 443, 461, 467, 479, 491, 499, 503, 509, 523, 541, 547, 557, 563, 577
Offset: 1

Views

Author

Max Alekseyev, Jan 14 2007

Keywords

Comments

Potential prime divisors of solutions to 2^m == 3 (mod m) (see A050259).
Minimal nonnegative solutions to 2^x == 3 (mod a(n)) are given in A127438.

Crossrefs

Cf. A050259, A123988 (complement in the primes).

Programs

  • PARI
    forprime(p=5,1000, g=znprimroot(p); u=znlog(Mod(2,p),g); v=znlog(Mod(3,p),g); if( v%u==0, print1(p,", "); ))

Extensions

Corrected by Max Alekseyev, Jun 08 2011
Corrected by Arkadiusz Wesolowski, Jan 12 2021

A001916 Primes p such that the congruence 2^x = 5 (mod p) is solvable.

Original entry on oeis.org

2, 3, 11, 13, 19, 29, 37, 41, 53, 59, 61, 67, 71, 79, 83, 101, 107, 131, 139, 149, 163, 173, 179, 181, 191, 197, 199, 211, 227, 239, 251, 269, 271, 293, 311, 317, 347, 349, 359, 373, 379, 389, 401, 409, 419, 421, 443, 449, 461, 467, 479, 491, 509, 521, 523, 541, 547, 557
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 64.
  • 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).

Links

Crossrefs

Cf. A001915.

Programs

  • Mathematica
    Select[Prime[Range[120]], MemberQ[Table[Mod[2^x-5, #], {x, 0, #}], 0]&] (* Jean-François Alcover, Aug 29 2011 *)

Extensions

Better description and more terms from David W. Wilson, Dec 12 2000
Description corrected by Joe K. Crump (joecr(AT)carolina.rr.com), Jan 17 2001

A123988 Primes p such that 2^x == 3 (mod p) has no solutions.

Original entry on oeis.org

3, 7, 17, 31, 41, 43, 73, 79, 89, 103, 109, 113, 127, 137, 151, 157, 199, 223, 229, 233, 241, 251, 257, 271, 277, 281, 283, 331, 337, 353, 367, 397, 401, 433, 439, 449, 457, 463, 487, 521, 569, 571, 593, 601, 607, 617, 631, 641, 673, 683, 691, 727, 733, 739, 751, 761, 809, 811, 823, 857, 881, 911
Offset: 1

Views

Author

Artur Jasinski, Nov 23 2006

Keywords

Comments

Such primes cannot divide solutions to 2^m == 3 (mod m) (see A050259).

Links

Crossrefs

Cf. A050259, A001915 (complement in the primes).

Programs

  • Magma
    lst:=[3]; for p in [5..911 by 2] do if IsPrime(p) then t:=0; e:=Ceiling(Log(2, p+1)); for x in [e..p-2] do if 2^x mod p eq 3 then t:=1; break; end if; end for; if t eq 0 then Append(~lst, p); end if; end if; end for; lst; // Arkadiusz Wesolowski, Jan 12 2021

Extensions

Edited by Max Alekseyev, Jan 14 2007
Corrected by Max Alekseyev, Jun 08 2011
Corrected by Arkadiusz Wesolowski, Jan 12 2021

A371811 Semiprimes q*p such that the congruence 2^x == q (mod p) is solvable, where q < p.

Original entry on oeis.org

6, 10, 14, 15, 22, 26, 33, 34, 38, 39, 46, 55, 57, 58, 62, 65, 69, 74, 77, 82, 86, 87, 91, 94, 95, 106, 111, 118, 122, 133, 134, 141, 142, 143, 145, 146, 158, 159, 166, 177, 178, 183, 185, 194, 201, 202, 203, 205, 206, 209, 213, 214, 218, 221, 226
Offset: 1

Views

Author

Juri-Stepan Gerasimov, Apr 06 2024

Keywords

Crossrefs

Subsequence of A006881. Apart from the first term, A100484 is a subsequence.
Cf. A001915, A001916.

Programs

  • Maple
    filter:= proc(n) local F,p,q,x;
  F:= ifactors(n)[2];
  if F[..,2] <> [1,1] then return false fi;
  p:= max(F[..,1]); q:= min(F[..,1]);
  [msolve(2^x = q, p)] <> []
end proc:
select(filter, [$6 .. 1000]); # Robert Israel, Apr 10 2024
  • Mathematica
    okQ[n_] := Module[{f, p, q, s},
   f = FactorInteger[n];
   If[f[[All, 2]] != {1, 1}, False,
   {q, p} = f[[All, 1]];
   s = Solve[Mod[2^x, p] == q, x, Integers];
   s != {}]];
Select[Range[6, 1000], okQ] (* Jean-François Alcover, May 03 2024 *)
  • PARI
    list(lim)=my(v=List()); forprime(p=3,lim\2, forprime(q=2,min(p-1,lim\p), if(znlog(q, Mod(2, p)) != [], listput(v,p*q)))); Set(v) \\ Charles R Greathouse IV, Apr 10 2024
  • Python
    from itertools import count, islice
from sympy import factorint, discrete_log
def A371811_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        if len(f) == 2 and max(f.values())==1:
            q, p = sorted(f.keys())
            try:
                discrete_log(p,q,2)
            except:
                continue
            yield n
A371811_list = list(islice(A371811_gen(),20)) # Chai Wah Wu, Apr 10 2024

Formula

Trivial bounds: n log n / log log n << a(n) << n log n. - Charles R Greathouse IV, Apr 10 2024

A290402 Primes congruent to {7, 17} mod 24.

Original entry on oeis.org

7, 17, 31, 41, 79, 89, 103, 113, 127, 137, 151, 199, 223, 233, 257, 271, 281, 353, 367, 401, 439, 449, 463, 487, 521, 569, 593, 607, 617, 631, 641, 727, 751, 761, 809, 823, 857, 881, 919, 929, 953, 967, 977, 991, 1039, 1049, 1063, 1087, 1097, 1193, 1217, 1231
Offset: 1

Views

Author

Arkadiusz Wesolowski, Aug 03 2017

Keywords

Comments

All these primes do not divide any number of the form 3*2^k - 1. Therefore, they are not in A001915.

Crossrefs

Cf. A001915, A107006, A107181.

Programs

  • Magma
    [p: p in PrimesUpTo(1231) | p mod 24 in {7, 17}];
  • Mathematica
    Select[Prime@Range[202], MemberQ[{7, 17}, Mod[#, 24]] &]

Formula

A107006 UNION A107181.
Showing 1-6 of 6 results.

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