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 1-3 of 3 results.

A301916 Primes which divide numbers of the form 3^k + 1.

Original entry on oeis.org

2, 5, 7, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 73, 79, 89, 97, 101, 103, 113, 127, 137, 139, 149, 151, 157, 163, 173, 193, 197, 199, 211, 223, 233, 241, 257, 269, 271, 281, 283, 293, 307, 317, 331, 337, 349, 353, 367, 373, 379, 389, 397, 401, 409, 439
Offset: 1

Views

Author

Luke W. Richards, Mar 28 2018

Keywords

Comments

This sequence can be used to factor P-1 values for prime candidates of the form 3^k+2, to aid with primality testing.
a(1) = 2 divides every number of the form 3^k+1. It is the only term with this property.
For k > 2, A000040(k) is a member if and only if A062117(k) is even. - Robert Israel, May 23 2018

Examples

			Every value of 3^k+1 is an even number, so 2 is in the sequence.
No values of 3^k+1 is ever a multiple of 3 for any integer k, so 3 is not in the sequence.
3^2+1 = 10, which is a multiple of 5, so 5 is in the sequence.

Links

Crossrefs

Cf. A000040, A034472, A062117, A301917, A320481.

Programs

  • Maple
    f:= p -> numtheory:-order(3,p)::even:
f(2):= true:
select(isprime and f, [2,seq(p,p=5..1000,2)]); # Robert Israel, May 23 2018
  • Mathematica
    Join[{2}, Select[Range[5, 1000, 2], PrimeQ[#] && EvenQ@ MultiplicativeOrder[3, #]&]] (* Jean-François Alcover, Feb 02 2023 *)
  • PARI
    isok(p)=if (p != 3, m = Mod(3, p); nb = znorder(m); for (k=1, nb, if (m^k == Mod(-1, p), return(1)););); return(0); \\ Michel Marcus, May 18 2018
  • PARI
    list(lim)=my(v=List([2]),t); forfactored(n=4,lim\1+1, if(n[2][,2]==[1]~, my(p=n[1],m=Mod(3,p)); for(k=2,znorder(m,t), m*=3; if(m==-1, listput(v,p); break))); t=n); Vec(v) \\ Charles R Greathouse IV, May 23 2018
  • PARI
    isok(p)=isprime(p)&&if(p<4,p==2,znorder(Mod(3,p))%2==0) \\ Jeppe Stig Nielsen, Jun 27 2020
  • PARI
    isok(p)=!isprime(p)&&return(0); p<4&&return(p==2); s=valuation(p-1,2); Mod(3,p)^((p-1)>>s)!=1 \\ Jeppe Stig Nielsen, Jun 27 2020

A045318 Primes p such that x^8 = 3 has no solution mod p.

Original entry on oeis.org

5, 7, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 73, 79, 89, 97, 101, 103, 113, 127, 137, 139, 149, 151, 157, 163, 173, 193, 197, 199, 211, 223, 233, 241, 257, 269, 271, 281, 283, 293, 307, 317, 331, 337, 349, 353, 367, 373, 379, 389, 397
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Complement of A045317 relative to A000040. - Vincenzo Librandi, Sep 19 2012

Links

Crossrefs

Cf. A000040, A045317, A301916, A320481.

Programs

  • Magma
    [p: p in PrimesUpTo(500) | not exists{x : x in ResidueClassRing(p) | x^8 eq 3} ]; // Vincenzo Librandi, Sep 19 2012
  • Mathematica
    ok[p_]:= Reduce[Mod[x^8 - 3, p] == 0, x, Integers] == False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 19 2012 *)

A301918 Primes which divide numbers of the form 3^k+3.

Original entry on oeis.org

2, 3, 5, 7, 17, 19, 29, 31, 37, 41, 43, 53, 61, 67, 73, 79, 89, 97, 101, 103, 113, 127, 137, 139, 149, 151, 157, 163, 173, 193, 197, 199, 211, 223, 233, 241, 257, 269, 271, 281, 283, 293, 307, 317, 331, 337, 349, 353, 367, 373, 379, 389, 397, 401, 409, 439
Offset: 1

Views

Author

Luke W. Richards, Mar 28 2018

Keywords

Comments

Union of {3} and A301916, because 3^k + 3 = 3*(3^(k-1) + 1). [Comment edited by Jeppe Stig Nielsen, Jul 04 2020.]
Can be used to factor P+1 values where P is a potential prime of the form 3^k+2.
Is this 2 and 3 with A045318? - David A. Corneth, May 04 2018
No, it is not. Primes like 769, 1297, ... are also here but not in A045318. See A320481 for the explanation. - Jeppe Stig Nielsen, Jun 27 2020

Examples

			All values of 3^k+3 are multiples of 2, so 2 is in the sequence.
3^4+3 = 84, which is a multiple of 7, so 7 is in the sequence.

Crossrefs

Cf. A045318, A301916, A301917, A301919.
Showing 1-3 of 3 results.

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

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