A268753 -id:A268753 - cp's OEIS frontend

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

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

Klaus Brockhaus, Jan 21 2001

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A000040, A049545, A268753.

[p: p in PrimesUpTo(4500) | forall{x: x in ResidueClassRing(p) | x^13 ne 2}]; // Bruno Berselli, Sep 12 2012

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  *)

A140444 Primes congruent to 1 (mod 14).

Original entry on oeis.org

29, 43, 71, 113, 127, 197, 211, 239, 281, 337, 379, 421, 449, 463, 491, 547, 617, 631, 659, 673, 701, 743, 757, 827, 883, 911, 953, 967, 1009, 1051, 1093, 1163, 1289, 1303, 1373, 1429, 1471, 1499, 1583, 1597, 1667, 1709, 1723, 1877, 1933, 2003, 2017, 2087

Offset: 1

Juri-Stepan Gerasimov, Jun 26 2008

From Federico Provvedi, May 24 2018: (Start)
Also primes congruent to 1 (mod 7).
For every prime p > 2, primes congruent to 1 (mod p) are also congruent to 1 (mod 2*p).
Conjecture: The monic polynomial P(x) = (x+1)^7/x - 1/x = ((x+1)^7-1)/x is irreducible but factorizable over Galois field (mod a(n)) with exactly 6 distinct irreducible factors of degree 1. Examples:
P(x) == (5 + x) (6 + x) (7 + x) (10 + x) (14 + x) (23 + x)    (mod 29)
P(x) == (3 + x) (9 + x) (23 + x) (28 + x) (33 + x) (40 + x)   (mod 43)
P(x) == (24 + x) (27 + x) (35 + x) (40 + x) (42 + x) (52 + x) (mod 71)
P(x) == (5 + x) (8 + x) (65 + x) (84 + x) (86 + x) (98 + x)   (mod 113)
... (End).
Primes in A131877. - Eric Chen, Jun 14 2018

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Jorma K. Merikoski, Pentti Haukkanen, and Timo Tossavainen, The congruence x^n = -a^n (mod m): Solvability and related OEIS sequences, Notes. Num. Theor. Disc. Math. (2024) Vol. 30, No. 3, 516-529. See p. 526.

Cf. A045437, A045458, A045471, A045473.
A090613 gives prime index.
Cf. A090614.
Cf. A131877.
Primes congruent to 1 (mod k): A000040 (k=1), A065091 (k=2), A002476 (k=3 and 6), A002144 (k=4), A030430 (k=5 and 10), this sequence (k=7 and 14), A007519 (k=8), A061237 (k=9 and 18), A141849 (k=11 and 22), A068228 (k=12), A268753 (k=13 and 26), A132230 (k=15 and 30), A094407 (k=16), A129484 (k=17 and 34), A141868 (k=19 and 38), A141881 (k=20), A124826 (k=21 and 42), A212374 (k=23 and 46), A107008 (k=24), A141927 (k=25 and 50), A141948 (k=27 and 54), A093359 (k=28), A141977 (k=29 and 58), A142005 (k=31 and 62), A133870 (k=32).

Filtered(Filtered([1..2300],n->n mod 14=1),IsPrime); # Muniru A Asiru, Jun 27 2018

[p: p in PrimesUpTo(3000)|p mod 14 in {1}]; // Vincenzo Librandi, Dec 18 2010

select(isprime,select(n->modp(n,14)=1,[$1..2300])); # Muniru A Asiru, Jun 27 2018

Select[Prime[Range[500]], Mod[#, 14] == 1 &]  (* Harvey P. Dale, Mar 21 2011 *)

is(n)=isprime(n) && n%14==1 \\ Charles R Greathouse IV, Jul 02 2016

a(n) ~ 6n log n. - Charles R Greathouse IV, Jul 02 2016

Simpler definition from N. J. A. Sloane, Jul 11 2008

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

Felix Fröhlich, Aug 08 2016

nonn

Intersection of A049545 and A268753.

These are the counterexamples mentioned in the Sep 12 2012 comment from Bruno Berselli in A059245.

			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

Cf. A049545, A059245, A268753.

Quiet@ Select[Prime@ Range[10^4], And[Mod[#, 13] == 1, IntegerQ@ PowerMod[2, 1/13, #]] &] (* Michael De Vlieger, Aug 10 2016 *)

forprime(p=1, 1e6, if(Mod(p, 13)==1 && ispower(Mod(2, p), 13), print1(p, ", ")))

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

A140444 Primes congruent to 1 (mod 14).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

GAP

Magma

Maple

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI