A213224 -id:A213224 - cp's OEIS frontend

A218357 Minimal order of degree-n irreducible polynomials over GF(5).

1, 3, 31, 13, 11, 7, 19531, 32, 19, 33, 12207031, 91, 305175781, 29, 181, 17, 409, 27, 191, 41, 379, 23, 8971, 224, 101, 5227, 109, 377, 59, 61, 1861, 128, 199, 1227, 211, 37, 149, 573, 79, 241, 2238236249, 43, 1644512641, 89, 209, 47, 177635683940025046467781066894531

Offset: 1

Alois P. Heinz, Oct 27 2012

nonn

a(n) < 5^n.

a(n) <= A143665(n). For prime n, a(n) = A143665(n). - Max Alekseyev, Apr 30 2022

Max Alekseyev, Table of n, a(n) for n = 1..520
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Cf. A212485, A213224.

with(numtheory):
M:= proc(n) M(n):= divisors(5^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..47);

M[n_] := M[n] = Divisors[5^n - 1] ~Complement~ U[n-1];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n-1]];
a[n_] := Min[M[n]];
Table[a[n], {n, 1, 47}] (* Jean-François Alcover, Mar 24 2017, translated from Maple *)

a(n) = min(M(n)) with M(n) = {d : d|(5^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A212485(n,1) = A213224(n,3).

A218358 Minimal order of degree-n irreducible polynomials over GF(7).

Original entry on oeis.org

1, 4, 9, 5, 2801, 36, 29, 64, 27, 11, 1123, 13, 16148168401, 113, 31, 17, 14009, 108, 419, 55, 261, 23, 47, 73, 2551, 53, 81, 145, 59, 99, 311, 256, 3631, 56036, 81229, 135, 223, 1676, 486643, 41, 83, 1017, 166003607842448777, 115, 837, 188, 13722816749522711

Offset: 1

Alois P. Heinz, Oct 27 2012

nonn

a(n) < 7^n.

Max Alekseyev, Table of n, a(n) for n = 1..430
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Cf. A057954, A074249, A212486, A213224.

with(numtheory):
M:= proc(n) M(n):= divisors(7^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..42);

M[n_] := M[n] = Divisors[7^n - 1]~Complement~U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n]~Union~U[n - 1]];
a[n_] := Min[M[n]];
Table[a[n], {n, 1, 47}] (* Jean-François Alcover, Oct 24 2022, after Alois P. Heinz *)

a(n) = min(M(n)) with M(n) = {d : d|(7^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A212486(n,1) = A213224(n,4).

A218356 Minimal order of degree-n irreducible polynomials over GF(3).

Original entry on oeis.org

1, 4, 13, 5, 11, 7, 1093, 32, 757, 44, 23, 35, 797161, 547, 143, 17, 1871, 19, 1597, 25, 14209, 67, 47, 224, 8951, 398581, 109, 29, 59, 31, 683, 128, 299, 103, 71, 95, 13097927, 2851, 169, 352, 83, 43, 431, 115, 181, 188, 1223, 97, 491, 151, 12853, 53, 107

Offset: 1

Alois P. Heinz, Oct 27 2012

nonn

a(n) < 3^n.

For n > 2, a(n) <= A143663(n). For odd prime n, a(n) = A143663(n). - Max Alekseyev, Apr 30 2022

Max Alekseyev, Table of n, a(n) for n = 1..796 (first 100 terms from Alois P. Heinz)

Cf. A143663, A212906, A213224, A235366.

M:= proc(n) M(n):= numtheory[divisors](3^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..60);

M[n_] := M[n] = Divisors[3^n - 1]~Complement~U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n]~Union~U[n - 1]];
a[n_] := Min[M[n]];
Table[a[n], {n, 1, 60}] (* Jean-François Alcover, Oct 24 2022, after Alois P. Heinz *)

a(n) = min(M(n)) with M(n) = {d : d|(3^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A212906(n,1) = A213224(n,2).

A212953 Minimal order of degree-n irreducible polynomials over GF(2).

Original entry on oeis.org

1, 3, 7, 5, 31, 9, 127, 17, 73, 11, 23, 13, 8191, 43, 151, 257, 131071, 19, 524287, 25, 49, 69, 47, 119, 601, 2731, 262657, 29, 233, 77, 2147483647, 65537, 161, 43691, 71, 37, 223, 174763, 79, 187, 13367, 147, 431, 115, 631, 141, 2351, 97, 4432676798593, 251

Offset: 1

Alois P. Heinz, Jun 01 2012

nonn

a(n) = smallest odd m such that A002326((m-1)/2) = n. - Thomas Ordowski, Feb 04 2014

For n > 1; n < a(n) < 2^n, wherein a(n) = n+1 iff n+1 is A001122 a prime with primitive root 2, or a(n) = 2^n-1 iff n is a Mersenne exponent A000043. - Thomas Ordowski, Feb 08 2014

			For n=4 the degree-4 irreducible polynomials p over GF(2) are 1+x+x^2+x^3+x^4, 1+x+x^4, 1+x^3+x^4. Their orders (i.e., the smallest integer e for which p divides x^e+1) are 5, 15, 15. (Example: (1+x+x^2+x^3+x^4) * (1+x) == x^5+1 (mod 2)). Thus the minimal order is 5 and a(4) = 5.

W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Springer 2004, Third Edition, 4.3 Factorization of Prime Ideals in Extensions. More About the Class Group (Theorem 4.33), 4.4 Notes to Chapter 4 (Theorem 4.40). - Regarding the first comment.

Max Alekseyev, Table of n, a(n) for n = 1..1236 (first 179 terms from Alois P. Heinz)
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Cf. A003060, A059912, A213224.

with(numtheory):
M:= proc(n) option remember;
      divisors(2^n-1) minus U(n-1)
    end:
U:= proc(n) option remember;
      `if`(n=0, {}, M(n) union U(n-1))
    end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..50);

M[n_] := M[n] = Divisors[2^n-1] ~Complement~ U[n-1];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n-1]];
a[n_] := Min[M[n]];
Array[a, 50] (* Jean-François Alcover, Mar 22 2017, translated from Maple *)

a(n) = min(M(n)) with M(n) = {d : d|(2^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A059912(n,1) = A213224(n,1).

A218359 Minimal order of degree-n irreducible polynomials over GF(11).

Original entry on oeis.org

1, 3, 7, 16, 25, 9, 43, 32, 1772893, 75, 15797, 13, 1093, 129, 175, 17, 50544702849929377, 27, 6115909044841454629, 400, 49, 23, 829, 224, 125, 53, 5559917315850179173, 29, 523, 31, 50159, 128, 661, 71707, 211, 351, 2591, 191, 79, 41, 83, 147, 1416258521793067

Offset: 1

Alois P. Heinz, Oct 27 2012

nonn

a(n) < 11^n.

Max Alekseyev, Table of n, a(n) for n = 1..330
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Cf. A213224, A218336, A274910.

with(numtheory):
M:= proc(n) M(n):= divisors(11^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..42);

M[n_] := M[n] = Divisors[11^n - 1]~Complement~U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n]~Union~U[n - 1]];
a[n_] := Min[M[n]];
Table[a[n], {n, 1, 43}] (* Jean-François Alcover, Oct 24 2022, after Alois P. Heinz *)

a(n) = min(M(n)) with M(n) = {d : d|(11^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A218336(n,1) = A213224(n,5).

A218360 Minimal order of degree-n irreducible polynomials over GF(13).

Original entry on oeis.org

1, 7, 9, 5, 30941, 63, 5229043, 32, 27, 11, 23, 45, 53, 29, 4651, 64, 103, 19, 12865927, 25, 43, 161, 1381, 288, 701, 371, 81, 145, 1973, 31, 311, 128, 207, 721, 211, 37, 1481, 90061489, 79, 41, 6740847065723, 261, 119627, 115, 181, 47, 183959, 576, 1667, 101

Offset: 1

Alois P. Heinz, Oct 27 2012

nonn

a(n) < 13^n.

Max Alekseyev, Table of n, a(n) for n = 1..416
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Cf. A213224, A218337.

with(numtheory):
M:= proc(n) M(n):= divisors(13^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..33);

M[n_] := M[n] = Divisors[13^n - 1]~Complement~U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n]~Union~U[n - 1]];
a[n_] := Min[M[n]];
Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Oct 24 2022, after Alois P. Heinz *)

a(n) = min(M(n)) with M(n) = {d : d|(13^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A218337(n,1) = A213224(n,6).

A218361 Minimal order of degree-n irreducible polynomials over GF(17).

Original entry on oeis.org

1, 3, 307, 5, 88741, 7, 25646167, 128, 19, 11, 2141993519227, 35, 212057, 22796593, 27243487, 256, 10949, 57, 229, 25, 43, 23, 47, 73, 2551, 53, 433, 5766433, 59, 31, 4093, 257, 67, 32847, 966211, 37, 149, 457, 157, 41, 83, 49, 1549, 89, 3691, 141

Offset: 1

Alois P. Heinz, Oct 27 2012

nonn

a(n) < 17^n.

Max Alekseyev, Table of n, a(n) for n = 1..372
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Cf. A213224, A218338.

with(numtheory):
M:= proc(n) M(n):= divisors(17^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..35);

M[n_] := M[n] = Divisors[17^n-1] ~Complement~ U[n-1];
U[n_] := U[n] = If[n == 0, {}, M[n] ~Union~ U[n-1]];
a[n_] := a[n] = Min[M[n]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 60}] (* Jean-François Alcover, Oct 21 2022, after Maple code *)

a(n) = min(M(n)) with M(n) = {d : d|(17^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A218338(n,1) = A213224(n,7).

A218362 Minimal order of degree-n irreducible polynomials over GF(19).

Original entry on oeis.org

1, 4, 27, 16, 151, 7, 701, 17, 81, 11, 104281, 13, 599, 197, 31, 64, 3044803, 199, 109912203092239643840221, 176, 18927, 23, 277, 119, 101, 131, 243, 29, 59, 61, 243270318891483838103593381595151809701, 97, 67, 12179212, 71, 37, 149, 108301, 79, 41, 10654507

Offset: 1

Alois P. Heinz, Oct 27 2012

nonn

a(n) < 19^n.

Max Alekseyev, Table of n, a(n) for n = 1..238
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Cf. A213224, A218339.

with(numtheory):
M:= proc(n) M(n):= divisors(19^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..28);

M[n_] := M[n] = Divisors[19^n - 1]~Complement~U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n]~Union~U[n - 1]];
a[n_] := Min[M[n]];
Table[a[n], {n, 1, 41}] (* Jean-François Alcover, Oct 24 2022, after Alois P. Heinz *)

a(n) = min(M(n)) with M(n) = {d : d|(19^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A218339(n,1) = A213224(n,8).

A218363 Minimal order of degree-n irreducible polynomials over GF(23).

Original entry on oeis.org

1, 3, 7, 5, 292561, 9, 29, 64, 19, 31, 121, 35, 47691619, 71, 2047927, 17, 103, 27, 2129, 25, 43, 363, 461, 448, 6551, 143074857, 4591, 145, 233, 151, 40888990028603, 193, 67, 239, 8484269, 73, 1925658337781, 6387, 333841333, 1600, 83, 129, 173, 605, 5558659

Offset: 1

Alois P. Heinz, Oct 27 2012

nonn

a(n) < 23^n.

Max Alekseyev, Table of n, a(n) for n = 1..306
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Cf. A213224, A218340.

with(numtheory):
M:= proc(n) M(n):= divisors(23^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..30);

M[n_] := M[n] = Divisors[23^n - 1]~Complement~U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n]~Union~U[n - 1]];
a[n_] := Min[M[n]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 45}] (* Jean-François Alcover, Oct 24 2022, after Alois P. Heinz *)

a(n) = min(M(n)) with M(n) = {d : d|(23^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A218340(n,1) = A213224(n,9).

A218364 Minimal order of degree-n irreducible polynomials over GF(29).

Original entry on oeis.org

1, 3, 13, 16, 732541, 9, 49, 32, 14437, 11, 23, 37, 521, 147, 181, 17, 3911, 19, 1386659, 176, 637, 69, 131327761273, 288, 151, 53, 52813, 784, 59, 99, 36767, 128, 299, 1973, 71, 304, 149, 16759, 169, 41, 83, 43, 173, 368, 2613097, 47, 283, 153, 197, 125, 103

Offset: 1

Alois P. Heinz, Oct 27 2012

nonn

a(n) < 29^n.

Max Alekseyev, Table of n, a(n) for n = 1..222
Eric Weisstein's World of Mathematics, Irreducible Polynomial
Eric Weisstein's World of Mathematics, Polynomial Order

Cf. A213224, A218341.

with(numtheory):
M:= proc(n) M(n):= divisors(29^n-1) minus U(n-1) end:
U:= proc(n) U(n):= `if`(n=0, {}, M(n) union U(n-1)) end:
a:= n-> min(M(n)[]):
seq(a(n), n=1..10);

M[n_] := M[n] = Divisors[29^n - 1]~Complement~U[n - 1];
U[n_] := U[n] = If[n == 0, {}, M[n]~Union~U[n - 1]];
a[n_] := Min[M[n]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 51}] (* Jean-François Alcover, Oct 24 2022, after Alois P. Heinz *)

a(n) = min(M(n)) with M(n) = {d : d|(29^n-1)} \ U(n-1) and U(n) = M(n) union U(n-1) for n>0, U(0) = {}.

a(n) = A218341(n,1) = A213224(n,10).

cp's OEIS Frontend

A218357 Minimal order of degree-n irreducible polynomials over GF(5).

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A218358 Minimal order of degree-n irreducible polynomials over GF(7).

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A218356 Minimal order of degree-n irreducible polynomials over GF(3).

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A212953 Minimal order of degree-n irreducible polynomials over GF(2).

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A218359 Minimal order of degree-n irreducible polynomials over GF(11).

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A218360 Minimal order of degree-n irreducible polynomials over GF(13).

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Maple

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A218361 Minimal order of degree-n irreducible polynomials over GF(17).

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