A108974 -id:A108974 - cp's OEIS frontend

A059912 Triangle T(n,k) of orders of n degree irreducible polynomials over GF(2) listed in ascending order, k=1..A059499(n).

1, 3, 7, 5, 15, 31, 9, 21, 63, 127, 17, 51, 85, 255, 73, 511, 11, 33, 93, 341, 1023, 23, 89, 2047, 13, 35, 39, 45, 65, 91, 105, 117, 195, 273, 315, 455, 585, 819, 1365, 4095, 8191, 43, 129, 381, 5461, 16383, 151, 217, 1057, 4681, 32767, 257, 771, 1285, 3855

Offset: 1

Vladeta Jovovic, Feb 09 2001

A permutation of the odd positive numbers; namely, order each odd number d by the multiplicative order of 2 modulo d (in case of a tie, smaller d go first). - Jeppe Stig Nielsen, Feb 13 2020

			There are 18 (cf. A001037) irreducible polynomials of degree 7 over GF(2) which all have order 127.
Triangle T(n,k) begins:
    1;
    3;
    7;
    5,   15;
   31;
    9,   21,  63;
  127;
   17,   51,  85, 255;
   73,  511;
   11,   33,  93, 341, 1023;
  ...

Alois P. Heinz, Rows n = 1..71, flattened
Eric Weisstein's World of Mathematics, Irreducible Polynomial
XIAO Gang, Polynomial order: computes the order of an irreducible polynomial over a finite field GF(p), WIMS.

Cf. A058943, A059478, A059499, A001037, A059913.
Cf. A212906, A212485, A212486.
Column k=1 of A212737.
Column k=1 gives: A212953.
Last elements of rows give: A000225.
Cf. A108974.

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:
T:= n-> sort([M(n)[]])[]:
seq(T(n), n=1..20);  # Alois P. Heinz, May 31 2012

m[n_] := m[n] = Complement[ Divisors[2^n - 1], u[n - 1]]; u[0] = {}; u[n_] := u[n] = Union[ m[n], u[n - 1]]; t[n_, k_] := m[n][[k]]; Flatten[ Table[t[n, k], {n, 1, 16}, {k, 1, Length[ m[n] ]}]] (* Jean-François Alcover, Jun 14 2012, after Alois P. Heinz *)

maxDegree=26;for(n=1,maxDegree,forstep(d=1,2^n,2,znorder(Mod(2,d))==n&&print1(d,", "))) \\ inefficient, Jeppe Stig Nielsen, Feb 13 2020

T(n,k) = k-th smallest element of M(n) = {d : d|(2^n-1)} \ U(n-1) with U(n) = M(n) union U(n-1) if n>0, U(0) = {}. - Alois P. Heinz, Jun 01 2012

A129733 List of primitive prime divisors of the numbers (3^k-1)/2 (A003462) for k>=2, in order of their occurrence.

Original entry on oeis.org

2, 13, 5, 11, 7, 1093, 41, 757, 61, 23, 3851, 73, 797161, 547, 4561, 17, 193, 1871, 34511, 19, 37, 1597, 363889, 1181, 368089, 67, 661, 47, 1001523179, 6481, 8951, 391151, 398581, 109, 433, 8209, 29, 16493, 59, 28537, 20381027, 31, 271, 683

Offset: 1

N. J. A. Sloane, May 13 2007

nonn

Read A003462 term-by-term, factorize each term, write down any primes not seen before.

Except for k=1, there is at least one primitive prime divisor for every k. - T. D. Noe, Mar 01 2010

Max Alekseyev, Primes for k <= 690 (primes for k <= 500 from T. D. Noe)
G. Everest et al., Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., 3 (1892), 265-284.

Cf. A003462, A076481, A028491.

If 3 is replaced with 2, we get A000225, A000043, A108974 respectively.

# produce sequence
s1:=(a,b,M)->[seq( (a^n-b^n)/(a-b),n=0..M)];
# find primes and their indices
s2:=proc(s) local t1,t2,i; t1:=[]; t2:=[];
for i from 1 to nops(s) do if isprime(s[i]) then
t1:=[op(t1),s[i]];
t2:=[op(t2),i-1]; fi; od; RETURN(t1,t2); end;
# get primitive prime divisors in order
s3:=proc(s) local t2,t3,i,j,k,np; t2:=[]; np:=0;
for i from 1 to nops(s) do t3:=ifactors(s[i])[2];
for j from 1 to nops(t3) do p := t3[j][1]; new:=1;
for k from 1 to np do if p = t2[k] then new:= -1; break; fi; od;
if new = 1 then np:=np+1; t2:=[op(t2),p]; fi; od; od;
RETURN(t2); end;

A129734 List of primitive prime divisors of the numbers 3^n-2^n (A001047) in their order of occurrence.

Original entry on oeis.org

5, 19, 13, 211, 7, 29, 71, 97, 1009, 11, 23, 331, 61, 53, 29927, 463, 3571, 17, 401, 129009091, 577, 1559, 745181, 4621, 43, 6217, 35839, 47, 2002867877, 5521, 101, 39756701, 79, 4057, 397760329, 369181, 68629840493971, 31, 241, 617671248800299, 3041, 14177

Offset: 1

N. J. A. Sloane, May 13 2007

nonn

Read A001047 term-by-term, factorize each term, write down any primes not seen before.

Amiram Eldar, Table of n, a(n) for n = 1..1893
Graham Everest, Shaun Stevens, Duncan Tamsett and Tom Ward, Primes generated by recurrence sequences, Amer. Math. Monthly, 114 (No. 5, 2007), 417-431.
K. Zsigmondy, Zur Theorie der Potenzreste, Monatsh. Math., 3 (1892), 265-284.

Cf. A001047, A058765, A057468; A003462, A076481, A028491, A129733; A000225, A004668, A000043, A108974.

a(41) and a(42) switched by Amiram Eldar, Jun 30 2023

A158895 A list of primes written in order of their first appearance in a table of prime factorizations of 2^k+1, k=1,2,... .

Original entry on oeis.org

3, 5, 17, 11, 13, 43, 257, 19, 41, 683, 241, 2731, 29, 113, 331, 65537, 43691, 37, 109, 174763, 61681, 5419, 397, 2113, 2796203, 97, 673, 251, 4051, 53, 157, 1613, 87211, 15790321, 59, 3033169, 61, 1321, 715827883

Offset: 1

Martin Griffiths, Mar 29 2009

This sequence has the property that if a(n) appears first in the table as a prime factor of 2^m+1 for some m then a(n)=2*k*m+1 for some k.

When, for some m, 2^m+1 has more than one prime factor appearing in the table for the first time, we adopt the convention of entering them in ascending order. For example, the entries ..., 29, 113, ... both arise from 2^14+1.

			2^1+1=3, 2^2+1=5, 2^3+1=3^2 and 2^4+1=17. Thus a(1)=3, a(2)=5 and a(3)=17, on noting that 2^3+1 contributes no new prime factors.

Harvey P. Dale and Charles R Greathouse IV, Table of n, a(n) for n = 1..4017 (first 650 terms from Dale)

Subsequence of A001269.

Cf. A002587, A066845, A108974.

DeleteDuplicates[Flatten[Table[Transpose[FactorInteger[2^k+1]][[1]],{k,50}]]] (* Harvey P. Dale, Mar 30 2014 *)

lista(n)=prs = Set(); for (k=1, n, f = factor(2^k+1); for (i=1, length(f~), onef = f[i,1]; if (! setsearch(prs, onef), print1(onef, ", "); prs = setunion(prs, Set(onef));););); \\ Michel Marcus, Apr 18 2013

G=1; for(n=1,500, g=gcd(f=2^n+1,G); while(g>1, g=gcd(g,f/=g)); f=factor(f)[,1]; if(#f, for(i=1,#f, print1(f[i]", ")); G*=factorback(f))) \\ Charles R Greathouse IV, Jan 03 2018

A283461 Second-largest prime factor of 2^n - 1, if composite, or 1 otherwise.

Original entry on oeis.org

1, 1, 3, 1, 3, 1, 5, 7, 11, 23, 7, 1, 43, 31, 17, 1, 19, 1, 31, 127, 89, 47, 17, 601, 2731, 73, 113, 1103, 151, 1, 257, 89, 43691, 127, 73, 223, 174763, 8191, 41, 13367, 337, 9719, 683, 631, 178481, 4513, 257, 127, 1801, 11119, 2731

Offset: 2

Charles R Greathouse IV, Mar 08 2017

nonn

For clarification: if the largest prime factor occurs more than once, then that prime factor is selected.

Charles R Greathouse IV, Table of n, a(n) for n = 2..1206

Cf. A005420, A193615, A006530, A000225, A108974.

a[n_] := If[PrimeQ[2^n-1], 1, Block[{f = FactorInteger[2^n-1]}, If[f[[-1, 2]] == 1, f[[-2, 1]], f[[-1, 1]]]]]; a /@ Range[2, 52] (* Giovanni Resta, Mar 08 2017 *)

a(n)=my(f=factor(2^n-1),t=#f~); if(f[t,2]>1, f[t,1], if(t>1, f[t-1,1], 1))

a(n) = A006530(A000225(n)/A005420(n)).

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A059912 Triangle T(n,k) of orders of n degree irreducible polynomials over GF(2) listed in ascending order, k=1..A059499(n).

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A129733 List of primitive prime divisors of the numbers (3^k-1)/2 (A003462) for k>=2, in order of their occurrence.

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A129734 List of primitive prime divisors of the numbers 3^n-2^n (A001047) in their order of occurrence.

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A158895 A list of primes written in order of their first appearance in a table of prime factorizations of 2^k+1, k=1,2,... .

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A283461 Second-largest prime factor of 2^n - 1, if composite, or 1 otherwise.

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