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 1-4 of 4 results.

A059245 Primes p such that x^13 = 2 has no solution mod p.

Original entry on oeis.org

53, 79, 131, 157, 313, 443, 521, 547, 599, 677, 859, 911, 937, 1093, 1171, 1223, 1249, 1301, 1327, 1483, 1613, 1847, 1873, 1951, 2003, 2029, 2081, 2237, 2341, 2393, 2549, 2731, 2861, 2887, 2939, 3121, 3251, 3329, 3407, 3433, 3511, 3719, 3797, 3823, 4057
Offset: 1

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Author

Klaus Brockhaus, Jan 21 2001

Keywords

Comments

Complement of A049545 relative to A000040.
Presumably this is the same as Primes congruent to 1 mod 13. - N. J. A. Sloane, Jul 11 2008
The smallest counterexample is 4421: 4421 == 1 (mod 13), but 162^13 == 2 (mod 4421), therefore this prime is not in the sequence. - Bruno Berselli, Sep 12 2012

Links

Crossrefs

Cf. A000040, A049545, A268753.

Programs

  • Magma
    [p: p in PrimesUpTo(4500) | forall{x: x in ResidueClassRing(p) | x^13 ne 2}]; // Bruno Berselli, Sep 12 2012
  • Mathematica
    Select[Prime[Range[PrimePi[5000]]], ! MemberQ[PowerMod[Range[#], 13, #], Mod[2, #]] &] (* T. D. Noe, Sep 12 2012 *)
ok[p_]:= Reduce[Mod[x^13 - 2, p] == 0, x, Integers] == False; Select[Prime[Range[600]], ok] (* Vincenzo Librandi, Sep 20 2012  *)

A127566 Primes p such that at least one of k-1, k+1 is prime, where k = absolute value of q^2 - p*r and p, q, r are consecutive primes.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 59, 61, 67, 71, 73, 83, 89, 97, 101, 103, 107, 113, 127, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 271, 277, 281, 283, 307, 311, 317, 331
Offset: 1

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Author

J. M. Bergot, Apr 02 2007

Keywords

Comments

k is always an even number.
Agrees with A049545 for the first 26 terms; first divergence is at 109.

Examples

			31, 37, 41 are consecutive primes, 31^2 - 37*41 = -556. 557 is prime, hence 31 is a term.
53, 59, 61 are consecutive primes, 59^2 - 53*61 = 248. Both 247 = 13*19 and 249 = 3*83 are composite, hence 53 is not in the sequence.

Links

Crossrefs

Cf. A049545.

Programs

  • Magma
    [ p: p in PrimesInInterval(2, 335) | IsPrime(k-1) or IsPrime(k+1) where k is Abs(q^2 - p*r) where r is NextPrime(q) where q is NextPrime(p) ]; /* Klaus Brockhaus, Apr 06 2007 */
  • Mathematica
    Transpose[Select[Partition[Prime[Range[70]],3,1],Or@@PrimeQ[Abs[ #[[2]]^2- #[[1]]*#[[3]]]+{1,-1}]&]][[1]] (* Harvey P. Dale, Oct 28 2013 *)

Extensions

Edited and extended by Klaus Brockhaus, Apr 06 2007

A164625 Primes p such that p+floor(p/2)+floor(p/3)+floor(p/5) is also prime.

Original entry on oeis.org

2, 3, 7, 19, 83, 89, 127, 137, 139, 181, 251, 257, 311, 317, 373, 379, 449, 491, 499, 503, 509, 673, 733, 797, 853, 857, 863, 919, 971, 983, 1033, 1039, 1049, 1093, 1151, 1201, 1217, 1399, 1453, 1579, 1583, 1627, 1697, 1741, 1871, 1933, 1993, 2129, 2237, 2281
Offset: 1

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Author

Vladimir Joseph Stephan Orlovsky, Aug 17 2009

Keywords

Examples

			For p=7, 7+3+2+1=13 is prime, which admits 7=a(4) to the sequence.
For p=19, 19+9+6+3=37 is prime, which puts 19=a(5) into the sequence.

Links

Crossrefs

Cf. A038874, A049545, A097933, A164624

Programs

  • Mathematica
    lst={};Do[p=Prime[n];If[PrimeQ[p+Floor[p/2]+Floor[p/3]+Floor[p/5]],AppendTo[lst, p]],{n,2*6!}];lst
Select[Prime[Range[350]],PrimeQ[Total[Floor[#/{2,3,5}]]+#]&] (* Harvey P. Dale, Feb 19 2012 *)

Extensions

Comments rephrased as examples by R. J. Mathar, Aug 20 2009

A275773 Primes p congruent to 1 modulo 13 such that x^13 = 2 has a solution modulo p.

Original entry on oeis.org

4421, 4733, 5669, 5981, 8581, 9413, 10453, 11597, 13963, 14327, 14951, 19267, 22699, 22907, 23557, 25117, 25819, 26417, 28627, 31799, 32579, 35491, 37441, 41549, 44773, 44851, 45553, 46619, 46957, 48179, 49297, 49921, 49999, 50207, 52859, 53639, 60217, 64403
Offset: 1

Views

Author

Felix Fröhlich, Aug 08 2016

Keywords

Comments

Intersection of A049545 and A268753.
These are the counterexamples mentioned in the Sep 12 2012 comment from Bruno Berselli in A059245.

Examples

			4421 is in the sequence since it is prime, it is congruent to 1 (mod 13), and x^13 == 2 (mod 4421) has the solution x = 162. - _Michael B. Porter_, Aug 26 2016

Crossrefs

Cf. A049545, A059245, A268753.

Programs

  • Mathematica
    Quiet@ Select[Prime@ Range[10^4], And[Mod[#, 13] == 1, IntegerQ@ PowerMod[2, 1/13, #]] &] (* Michael De Vlieger, Aug 10 2016 *)
  • PARI
    forprime(p=1, 1e6, if(Mod(p, 13)==1 && ispower(Mod(2, p), 13), print1(p, ", ")))
Showing 1-4 of 4 results.

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