A059245 -id:A059245 - cp's OEIS frontend

A049545 Primes p such that x^13 = 2 has a solution mod p.

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, 109, 113, 127, 137, 139, 149, 151, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293

Offset: 1

N. J. A. Sloane

Complement of A059245 relative to A000040. - Vincenzo Librandi, Sep 13 2012

			0^13 == 2 (mod 2). 2^13 == 2 (mod 3). 2^13 == 2 (mod 5). 2^13 == 2 (mod 7). 7^13 == 2 (mod 11). 2^13 == 2 (mod 13). 15^13 == 2 (mod 17). 14^13 == 2 (mod 19). 18^13 == 2 (mod 23). 14^13 == 2 (mod 29). 4^13 == 2 (mod 31). 20^13 == 2 (mod 37). 36^13 == 2 (mod 41). 22^13 == 2 (mod 43). - _R. J. Mathar_, Jul 20 2025

R. J. Mathar, Table of n, a(n) for n = 1..1000
Index entries for related sequences

Cf. A000040, A059245.

[p: p in PrimesUpTo(400) | exists(t){x : x in ResidueClassRing(p) | x^13 eq 2}]; // Vincenzo Librandi, Sep 13 2012

ok[p_]:= Reduce[Mod[x^13- 2, p] == 0, x, Integers]=!=False; Select[Prime[Range[200]], ok] (* Vincenzo Librandi, Sep 13 2012 *)

A268753 Primes congruent to 1 mod 13.

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, 4421, 4447, 4603, 4733, 4759, 4889, 4967, 4993, 5227, 5279

Offset: 1

Alexei Kourbatov, Feb 12 2016

The first 45 terms, up to 4057, coincide with A059245. Then a(46)=4421 occurs in this sequence, while A059245(46)=4447.

			53 is the first prime of the form 13k + 1, therefore a(1)=53.

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A059245 (x^13 = 2 has no solution mod prime p).

[p: p in PrimesUpTo(5300) | p mod 13 in {1} ]; // Vincenzo Librandi, Feb 13 2016

Select[Prime@ Range@ 700, Mod[#, 13] == 1 &] (* Michael De Vlieger, Feb 12 2016 *)

forprime(p=2, 1e4, if(p%13==1, print1(p", ")))

forprimestep(p=53,1e4,26,print1(p", ")) \\ Charles R Greathouse IV, Mar 11 2020

a(n) ~ 12n log n. - Charles R Greathouse IV, Mar 11 2020

A140371 Primes of the form 26k + 7.

Original entry on oeis.org

7, 59, 137, 163, 241, 293, 397, 449, 631, 683, 709, 761, 787, 839, 1021, 1151, 1229, 1307, 1489, 1567, 1619, 1697, 1723, 1801, 1879, 1931, 2087, 2113, 2243, 2269, 2347, 2399, 2477, 2503, 2633, 2659, 2711, 2789, 2971, 3023, 3049, 3257, 3361, 3413, 3491, 3517

Offset: 1

Juri-Stepan Gerasimov, Jun 14 2008

nonn

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A059245, A100202, A102732, A093191, A093350, A092178.

[a: n in [0..250]|IsPrime(a) where a is 26*n+7] // Vincenzo Librandi, Dec 18 2010

Select[Range[7, 30000, 26], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)

Edited by R. J. Mathar, Jun 16 2008

A140373 Primes of the form 26*n+11.

Original entry on oeis.org

11, 37, 89, 167, 193, 271, 349, 401, 479, 557, 661, 739, 947, 1051, 1103, 1129, 1181, 1259, 1493, 1571, 1597, 1753, 1831, 1987, 2039, 2143, 2221, 2273, 2351, 2377, 2663, 2689, 2741, 2767, 2819, 2897, 3001, 3079, 3209, 3313, 3391, 3469, 3547, 3677, 3833

Offset: 1

Juri-Stepan Gerasimov, Jun 14 2008

Also primes of the form 13*n+11. - N. J. A. Sloane, Jul 11 2008

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A059245, A100202, A102732, A093191, A093350, A092178, A111369 (gives 2*n).

[a: n in [0..250]|IsPrime(a) where a is 26*n+11] // Vincenzo Librandi, Dec 18 2010

Select[Range[11, 50000, 13], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Jun 13 2011 *)

Edited by R. J. Mathar, Jun 16 2008

A140375 Primes of the form 26n+23.

Original entry on oeis.org

23, 101, 127, 179, 257, 283, 439, 491, 569, 647, 673, 751, 829, 881, 907, 1063, 1193, 1297, 1427, 1453, 1531, 1583, 1609, 1973, 1999, 2129, 2207, 2311, 2389, 2441, 2467, 2753, 2857, 2909, 3169, 3221, 3299, 3533, 3559, 3637, 3767, 3793, 3923, 4001, 4027

Offset: 1

Juri-Stepan Gerasimov, Jun 14 2008

Also primes congruent to 10 mod 13. - N. J. A. Sloane, Jul 11 2008

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

Cf. A000040, A059245, A100202, A102732, A093191, A093350, A092178.

[a: n in [0..250]|IsPrime(a) where a is 26*n+23]; // Vincenzo Librandi, Dec 18 2010

Select[Range[10, 50000, 13], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Jun 13 2011 *)
Select[Prime[Range[200]],MemberQ[{10},Mod[#,13]]&] (* Vincenzo Librandi, Aug 15 2012 *)

Edited by R. J. Mathar, Jun 16 2008

A140372 Primes of the form 26k + 9.

Original entry on oeis.org

61, 113, 139, 191, 269, 347, 373, 503, 607, 659, 919, 971, 997, 1049, 1153, 1231, 1283, 1361, 1439, 1543, 1621, 1699, 1777, 1907, 1933, 2011, 2063, 2089, 2141, 2297, 2531, 2557, 2609, 2687, 2713, 2791, 2843, 2999, 3181, 3259, 3389, 3467, 3571, 3623, 3701

Offset: 1

Juri-Stepan Gerasimov, Jun 14 2008

Also primes of the form 13k + 9. - N. J. A. Sloane, Jul 11 2008

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A059245, A100202, A102732, A093191, A093350, A092178.

[a: n in [0..250]|IsPrime(a) where a is 26*n+9] // Vincenzo Librandi, Dec 18 2010

Select[Range[9, 30000, 26], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Jun 21 2011 *)

Edited by R. J. Mathar, Jun 16 2008

A140374 Primes of the form 26k + 15.

Original entry on oeis.org

41, 67, 197, 223, 353, 379, 431, 457, 509, 587, 613, 691, 743, 769, 821, 977, 1237, 1289, 1367, 1471, 1523, 1549, 1601, 1627, 1783, 1861, 1913, 2017, 2069, 2251, 2381, 2459, 2693, 2719, 2797, 2927, 2953, 3083, 3109, 3187, 3343, 3499, 3733, 3863, 3889

Offset: 1

Juri-Stepan Gerasimov, Jun 14 2008

nonn

Cf. A059245, A100202, A102732, A093191, A093350, A092178.

[a: n in [0..250]|IsPrime(a) where a is 26*n+15] // Vincenzo Librandi, Dec 18 2010

Select[Range[15, 20000, 26], PrimeQ] (* Vladimir Joseph Stephan Orlovsky, Jun 18 2011 *)

Edited by R. J. Mathar, Jun 16 2008

A140376 Nonprimes of the form 26n+1.

Original entry on oeis.org

1, 27, 105, 183, 209, 235, 261, 287, 339, 365, 391, 417, 469, 495, 573, 625, 651, 703, 729, 755, 781, 807, 833, 885, 963, 989, 1015, 1041, 1067, 1119, 1145, 1197, 1275, 1353, 1379, 1405, 1431, 1457, 1509, 1535, 1561, 1587, 1639, 1665, 1691, 1717, 1743

Offset: 1

Juri-Stepan Gerasimov, Jun 14 2008

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

Cf. A059245, A100202, A102732, A093191, A093350, A092178.

[a: n in [0..80] | not IsPrime(a) where a is 26*n+1]; // Vincenzo Librandi, Mar 22 2014

Select[26 Range[0, 100] + 1, ! PrimeQ@# &] (* Vincenzo Librandi, Mar 22 2014 *)

Edited by R. J. Mathar, Jun 16 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

A049545 Primes p such that x^13 = 2 has a solution mod p.

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A268753 Primes congruent to 1 mod 13.

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A140371 Primes of the form 26k + 7.

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A140373 Primes of the form 26*n+11.

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A140375 Primes of the form 26n+23.

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A140372 Primes of the form 26k + 9.

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A140374 Primes of the form 26k + 15.

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A140376 Nonprimes of the form 26n+1.

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